(a) First we observe that the map
\(\Upsilon \colon  \mathcal {H}\rightarrow \mathcal {H}^{\prime }, v \mapsto \alpha _v\)  is an antilinear
isometry and that the contragredient representation
\(\pi ^*(g)\alpha := \alpha \circ \pi (g)^{-1}\)  is also unitary.
Next we recall that the smooth action of \(G\)  on
\(M := G/H\)  defines a homomorphism
\({\mathfrak {g}}\rightarrow \mathcal {V}_{\cal O}(M)\) , the Lie algebra of holomorphic vector fields.
Since \(M\)  is complex, \(\mathcal {V}_{\cal O}(M)\)  is a complex
Lie algebra, and, for each \(p \in M\) , the subspace
\(\lbrace  X \in \mathcal {V}_{\cal O}(M) \colon  X(p) = 0\rbrace \)  is a complex subalgebra.
This proves that \({\mathfrak {p}}\)  is a Lie subalgebra of \({\mathfrak {g}}_, and its closednessfollows from the continuity of the map \) gTx0(M) g/h\(.\) (b) Since \(v\)  is a \(\overline{{\mathfrak {p}}}\) -eigenvector, there exists a continuous
linear functional \(\lambda \colon  {\mathfrak {p}}\rightarrow  with\)d(z) v = (z) v\( for\) z p\(, so that we have for\) w H\( and \) z p\( the relation\)\( \alpha _v(z.w)= \langle {\tt d}\pi (z)w, v \rangle = - \langle w, {\tt d}\pi (\overline{z})v \rangle = - \lambda (z) \alpha _v(w). \)\(We conclude that \) v\( is an \) p\(-eigenvector. This implies thatthe tangent map\)\( T_{x_0}(\eta ) \colon  T_{x_0}(M) \rightarrow T_{[\alpha _v]}({\mathbb {P}}(\mathcal {H}^{\prime })) \)\(is complex linear, i.e., compatible with the respective complexstructures. Since \) M\( is homogeneous, \) T()\( is complexlinear on each tangent space, and this means that \)\( is holomorphic.Therefore \) *LH'\( is a holomorphic line bundle over\) M\( and we obtain a \) G\(-equivariant pullback map\) H(LH') (*LH')\(.Its kernel consists of all those sectionsvanishing on \) (M)\(, which corresponds to the elements\) w ((G)v)\(. In particular, this map is injective if\) v\( is a cyclic vector.\) That \((\pi , \mathcal {H})\)  is irreducible if \(G\)  is connected follows from
[1]}.
(c) If \(x \in W_\pi \) , then \(i{\tt d}\pi (x)\)  is bounded from above,
so that, so that \((z,\alpha ) \mapsto \alpha \circ e^{z {\tt d}\pi (x)}\) 
defines a continuous action of \(+\)  on \(\mathcal {H}^{\prime }\)  which is holomorphic
on \(+^0\)  and satisfies \((t,\alpha ) \mapsto \pi ^*(\exp _G(-tx))\alpha \) 
(cf. Example REF ).
As \(\eta \)  is \(G\) -equivariant and holomorphic,
it is also equivariant with respect to the \(+\) -actions
on \(M\)  and the holomorphic action on \({\mathbb {P}}(\mathcal {H}^{\prime })\)  by
\((z,[\alpha ]) \mapsto [\alpha \circ e^{z {\tt d}\pi (x)}]\) .
Therefore the \(+\)  actions on \(M\)  and on \(\mathcal {H}^{\prime }\)  combine to a holomorphic
action of \(+\)  on \(\eta ^*{\mathbb {L}}_{\mathcal {H}^{\prime }}\) . Now we combine
Theorem REF  with
Remark REF  to obtain (c). Here we only have to
observe that the realization of \(\mathcal {H}\)  by holomorphic sections
of \({\mathbb {L}}_{\mathcal {H}^{\prime }}\)  leads on the dual bundle, whose complement
of the zero-section can be identified with \(\mathcal {H}^{\prime }\setminus \lbrace 0\rbrace \) ,
to the evaluation functional \(K_{\alpha _v} = v \in \mathcal {H}\) 
(cf. Example REF ).
\(\Box \)
Example 5.12  
An important special case where the
requirements of Theorem REF  are satisfied
occurs if the \(G\) -action on the complex manifold
\(M\)  extends to a holomorphic action of a complex
Lie group \(G_G\)  with holomorphic exponential function.
If \((\pi , \mathcal {H})\)  is an irreducible representation of a compact
Lie group \(G\) , then it is finite dimensional and in particular
bounded, so that \(\pi \)  extends to a holomorphic representation
\(\widehat{\pi }\colon  G_\mathop {{\rm GL}}\nolimits (\mathcal {H})\) . Since the highest weight orbit
\(G[v_\lambda ]\)  is a compact complex manifold, it is also invariant
under the \(G_-action on \) P(H)\(, and Theorem~\ref {thm:5.7},applied to the maximal torus \) H G\(,implies that \) I= conv(O)\( for\) O= (G[v])\(.\)
Remark 5.13  (a) If \(G\)  is finite dimensional, then Theorem REF  applies
to all irreducible semibounded representations
([1]}), where \(G/H\)  is the
highest weight orbit \(G[v_\lambda ] \subseteq {\mathbb {P}}(\mathcal {H}_\lambda )\) , and this eventually leads to
\(I_{\pi _\lambda } = \overline{\mathop {{\rm conv}}\nolimits }(\mathcal {O}_{-i\lambda })\) .
(b) We have already seen in (REF ) that,
for finite dimensional groups,
the highest weight orbit \(G[v_\lambda ] \subseteq {\mathbb {P}}(\mathcal {H}_\lambda )\) 
can be characterized as the inverse image of the set
\(\mathop {{\rm Ext}}\nolimits (I_{\pi _\lambda })\)  of extreme points of the momentum set
(Remark REF ).
For the special case where \(G\)  is a compact group, one finds in
[3]} a different characterization of the highest weight
orbit as the set of all those elements for which
the subrepresentation of \(\mathop {{\rm Sym}}\nolimits ^2(\mathcal {H}_\lambda )\)  generated by
\(v \otimes v\)  is irreducible. It would be interesting to see if a
similar result holds in other situations.
If \(G\)  is finite dimensional, for any
\([v] \in G[v_\lambda ]\)  the orbit of \([v \otimes v]\)  is
a complex manifold (actually a holomorphic image of
\(G[v_\lambda ])\) , and from that one can derive that the cyclic
representation it generates is irreducible ([1]}).
Another interesting characterization of the highest weight orbit
of an irreducible representation of a compact Lie group
is given in [5]}. Starting with an orthonormal basis
\(i F_1,\ldots , i F_n\) 
of the Lie algebra \({\mathfrak {g}}\)  with respect to an invariant scalar
product, one defines the invariant dispersion of a state \([v]\) 
by
\( (\Delta F)^2 = \Big \langle \sum _r (F_r - \langle F_r \rangle )^2 \Big \rangle ,\quad \mbox{ where } \quad \langle A \rangle = \frac{\langle Av,v\rangle }{\langle v,v\rangle } \quad \mbox{ for } \quad A = A^*. \) 
If \(C := \sum _r F_r^2\)  is the corresponding Casimir operator,
which acts on \(\mathcal {H}\)  as a multiple \(c \mathbf {1}\) 
of the identity, one easily finds that
\( (\Delta F)^2 = c - \Vert \Phi _\pi ([v])\Vert ^2, \) 
which is minimal if \(\Vert \Phi _\pi ([v])\Vert \)  is maximal.
As \(I_\pi = \mathop {{\rm conv}}\nolimits (\mathcal {O}_\pi )\)  holds for a coadjoint orbit \(\mathcal {O}_\pi \) ,
and the scalar product on
\({\mathfrak {g}}\) , resp., \({\mathfrak {g}}^{\prime }\)  is invariant, the orbit \(\mathcal {O}_\pi \)  is contained
in a sphere. Therefore the invariant dispersion \((\Delta F)^2\) 
is minimal in a state \([v]\) 
if and only if \(\Phi _\pi ([v]) \in \mathcal {O}_\pi = \mathop {{\rm Ext}}\nolimits (I_\pi )\) .

Invariant cones in Lie algebras
In this section we take a closer look at important examples
of invariant convex cones in Lie algebras.
Invariant cones in unitary Lie algebras
Examples 6.1  
If \(\mathcal {H}\)  is a complex Hilbert space, then the
Lie algebra \({\mathfrak {u}}(\mathcal {H})\)  of skew-hermitian bounded operators
contains the open invariant cone
\( C_{{\mathfrak {u}}(\mathcal {H})} := \lbrace  x \in {\mathfrak {u}}(\mathcal {H}) \colon  i x <\!\!< 0 \rbrace  \) 
(cf. Example REF ).

More generally, for any unital \(C^*\) -algebra \(\mathcal {A}\) , the
Banach–Lie algebra \({\mathfrak {u}}(\mathcal {A}) = \lbrace  x \in \mathcal {A}\colon  x^* = -x \rbrace \)  contains
the open invariant cone
\( C_{{\mathfrak {u}}(\mathcal {A})} := \lbrace  x \in {\mathfrak {u}}(\mathcal {A}) \colon  i x <\!\!< 0 \rbrace . \) 


Invariant cones in symplectic Lie algebras
Definition 6.2  
If \(V\)  is a Banach space and
\(q \colon  V\rightarrow {\mathbb {R}}\)  a continuous quadratic form, then we
say that \(q\)  is strongly positive definite, written
\(q >\!\!> 0\) , if \(\sqrt{q}\)  defines a Hilbert norm on \(V\) .
In particular we are asking for \(V\)  to be complete with respect
to this norm.

Now let \((V,\omega )\)  be a strongly symplectic Banach space
(cf. Remark REF )
and \(\mathfrak {sp}(V,\omega ) \subseteq \mathop {{\mathfrak {gl} }}\nolimits (V)\)  be the corresponding symplectic
Lie algebra. We associate to each \(X \in \mathfrak {sp}(V,\omega )\)  the
quadratic Hamiltonian \(H_X(v) := \frac{1}{2}\omega (Xv,v)\)  and obtain
an open invariant cone by 
\( W_{\mathfrak {sp}(V,\omega )} := \lbrace  X \in \mathfrak {sp}(V,\omega ) \colon  H_X >\!\!> 0 \rbrace  \) 
which 
is non-empty if and only if \(V\)  is topologically isomorphic
to a real Hilbert space which carries a complex Hilbert space
structure \(\langle \cdot ,\cdot \rangle \)  with
\( \omega (v,w) =\mathop {{\rm Im}}\nolimits \langle v, w\rangle \quad \mbox{ for } \quad v,w \in V \) 
(cf. Proposition REF , [6]}).

Definition 6.3  
Actually \(\mathfrak {sp}(V,\omega )\)  is a Lie subalgebra of the semidirect product
\( \mathfrak {hsp}(V,\omega ) := \mathfrak {heis}(V,\omega ) \rtimes \mathfrak {sp}(V,\omega ), \) 
 where \(\mathfrak {heis}(V,\omega ) = {\mathbb {R}}\oplus _\omega V\)  is the Heisenberg algebra
associated to \((V,\omega )\) , with the bracket
\( [(z,v), (z^{\prime },v^{\prime })] := (\omega (v,v^{\prime }), 0). \) 
Since every continuous linear functional on \(V\)  is of the form
\(i_x\omega \) , the discussion in Remark REF 
implies that
\(\mathfrak {hsp}(V,\omega )\)  can be identified with the space of
continuous polynomials of degree \(\le 2\)  on \(V\) , endowed with the
Poisson bracket
\( \lbrace f,g\rbrace  = \omega (X_g, X_f)\quad \mbox{ where }\quad i_{X_f} \omega = {\tt d}f. \) 
Let \(\mathop {{\rm Heis}}\nolimits (V,\omega ) := {\mathbb {R}}\times V\)  be the Heisenberg group
of \((V,\omega )\) with the multiplication
\( (t,v) (t^{\prime },v^{\prime }) := (t + t^{\prime } + {\textstyle {\frac{1}{2}}}\omega (v,v^{\prime }), v + v^{\prime }). \) 
Then the Jacobi group

\(\mathop {\rm HSp{}}\nolimits (V,\omega ) := \mathop {{\rm Heis}}\nolimits (V,\omega ) \rtimes \mathop {{\rm Sp}}\nolimits (V,\omega )\)  acts by
\( \sigma \big ((c,w,g), v\big ) := w + gv\)  on \(V\)  and the
corresponding derived action is given by
\(\dot{\sigma }(z,w,x)(v) = - w - Xv\) , so that
\( (i_{\dot{\sigma }(z,w,x)}\omega )_v = -\omega (w + Xv, \cdot )= - i_{w + Xv}\omega . \) 
Therefore this action is Hamiltonian with equivariant momentum map
\( \Phi \colon  \mathcal {H}\rightarrow \mathfrak {hsp}(V,\omega )^{\prime }, \quad \Phi (v)(c,w,A) := - c - \omega (w,v) - {\textstyle {\frac{1}{2}}}\omega (Av,v) \) 
(cf. [1]}, where we use a different sign
convention).

Lemma 6.4   The convex cone
\(W_{\mathfrak {hsp}(V,\omega )}^+&:= \lbrace  (c,v,A) \in \mathfrak {hsp}(V,\omega ) \colon  (\forall v \in V)c + \omega (x,v) + H_A(v) > 0, H_A >\!\!> 0 \rbrace \) 

is open and invariant in the Banach–Lie algebra \(\mathfrak {hsp}(V,\omega )\) .
It is contained in the larger open invariant cone
\(W_{\mathfrak {hsp}(V,\omega )}&:= \lbrace  (c,v,A) \in \mathfrak {hsp}(V,\omega ) \colon  H_A >\!\!> 0 \rbrace .\) 


Proof.   
It is clear that \(W := W_{\mathfrak {hsp}(V,\omega )}^+\)  is an invariant
convex cone. It remains to show that it is open.
If \(f(v) = c + \omega (x,v) + \frac{1}{2}\omega (Av,v)\)  is such that
\(H_A\)  is strictly positive, then
\({\tt d}f(v) = i_x\omega + i_{Av}\omega ,\) 
which vanishes if and only if \(v = - A^{-1}x\) . It follows in particular,
that each such function has a unique minimal value which is given by
\( f(-A^{-1}x) = c - \omega (x,A^{-1}x) + \frac{1}{2} \omega (x,A^{-1}x)= c - \frac{1}{2} \omega (x,A^{-1}x). \) 
Therefore the condition \(f > 0\)  is equivalent to
\(c > \frac{1}{2} \omega (x,A^{-1}x),\) 
showing that \(W\)  is indeed open in \(\mathfrak {hsp}(V,\omega )\) .
\(\Box \)
From now on we assume that \(\mathcal {H}\)  is a complex Hilbert space,
\(V = \mathcal {H}_{\mathbb {R}}\)  is the underlying real Banach space, and
\(\omega (v,w) := \mathop {{\rm Im}}\nolimits \langle v,w \rangle \)  is the corresponding symplectic form.
Then \(Iv = iv\)  is a complex structure on \(\mathcal {H}_{\mathbb {R}}\)  leaving \(\omega \) 
invariant. It satisfies
\(\omega (Iv,v) = \mathop {{\rm Im}}\nolimits \langle iv,v\rangle = \Vert v\Vert ^2.\) 
Formalizing this property leads to:
Definition 6.5 
We call a real linear complex structure \(J \colon  \mathcal {H}\rightarrow \mathcal {H}\) 
\(\omega \) -positive if \(\omega (Jv,w)\)  is symmetric and
positive definite and write \({\mathcal {I}}_\omega \)  for the set of \(\omega \) -positive
complex structures on \(\mathcal {H}\) .

The following lemma is well known for the finite dimensional
case, but it carries over to infinite dimensional Hilbert
spaces. It implies in particular that
\({\mathcal {I}}_\omega = \mathcal {O}_I\)  is an adjoint orbit of the Lie algebra \(\mathfrak {sp}(\mathcal {H})\) .
Lemma 6.6  The following assertions hold:
(i)
\(\mathop {{\rm Sp}}\nolimits (\mathcal {H}) \cap \mathfrak {sp}(\mathcal {H})= \lbrace  g \in \mathop {{\rm GL}}\nolimits (\mathcal {H}_{\mathbb {R}}) \colon  I g^\top I = - g^{-1} = g \rbrace \) 
is the set of complex structures \(J\)  on \(\mathcal {H}\)  for which
\(\omega (Jv,w)\)  is symmetric.
(ii)
\({\mathcal {I}}_\omega = I e^{{\mathfrak {p}}}\)  for
\({\mathfrak {p}}:= \lbrace  x \in \mathfrak {sp}(\mathcal {H}) \colon  Ix = - xI\rbrace = \lbrace  x \in \mathfrak {sp}(\mathcal {H}) \colon  x^\top = x\rbrace .\)
(iii)
The conjugation action of \(\mathop {{\rm Sp}}\nolimits (\mathcal {H})\)  on
\({\mathcal {I}}_\omega \)  leads to a diffeomorphism \({\mathcal {I}}_\omega \cong \mathop {{\rm Sp}}\nolimits (\mathcal {H})/\mathop {\rm U{}}\nolimits (\mathcal {H})\) .
(iv)
If \(A \in \mathfrak {sp}(\mathcal {H})\)  is such that \(H_A >\!\!> 0\) , then there exists a
unique \(\omega \) -positive complex structure \(J\)  on \(\mathcal {H}\)  commuting with \(A\) .
(v)
\({\mathcal {I}}_\omega ^{\rm res} := \mathop {{\rm Ad}}\nolimits (\mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H}))I= \lbrace  J \in {\mathcal {I}}_\omega \colon  \Vert I - J \Vert _2 < \infty \rbrace = {\mathcal {I}}_\omega \cap \mathfrak {sp}_{\rm res}(\mathcal {H})\) .


Proof.   
(i) That \(\omega (Jv,w)\)  is symmetric is equivalent to
\(J \in \mathfrak {sp}(\mathcal {H})\) , and as \(J^{-1} = -J\)  characterizes complex structures,
(i) follows.
(ii) We have seen in (REF )
that \(I\)  is \(\omega \) -positive.
Let \(J\)  be another \(\omega \) -positive complex structure.
Writing \(J = u e^x\)  according to the polar
decomposition of \(\mathop {{\rm Sp}}\nolimits (\mathcal {H})\)  with
\(u \in \mathop {\rm U{}}\nolimits (\mathcal {H})\)  and \(x\in {\mathfrak {p}}\)  ([8]}),
we see that \(J^2 = -\mathbf {1}\)  is equivalent to \(u^2 = -\mathbf {1}\) 
(\(u\)  is a complex structure) and \(ux=-xu\)  (\(x\)  is antilinear
with respect to \(u\) ). If this is the case, then
\(\omega (Jv,v) = \omega (u e^x v, v)= \omega (e^{-x/2}u e^{x/2}v,v)= \omega (u e^{x/2}v,e^{x/2}v) \) 
shows that \(J\)  is \(\omega \) -positive if and only if
\(u\)  has this property. Since \(u\)  is complex linear,
\(\mathcal {H}\)  decomposes into \(\pm i\) -eigenspaces
\(\mathcal {H}_\pm \)  of \(u\) . Then \(\omega (uv,v)\)  is positive definite on
\(\mathcal {H}_+\)  and negative definite on \(\mathcal {H}_-\) , so that the
\(\omega \) -positivity implies \(u = I\) . This proves (ii).
(iii) For \(g = u e^x \in \mathop {{\rm Sp}}\nolimits (\mathcal {H})\) , the relation
\(g^{-1}Ig = e^{-x} I e^x = I e^{2x}\) 
shows that \(\mathop {{\rm Sp}}\nolimits (\mathcal {H})\)  acts transitively on
\({\mathcal {I}}_\omega \) . As \(\mathop {\rm U{}}\nolimits (\mathcal {H})\)  is the stabilizer of \(I\) ,
(iii) follows from the smoothness of the map
\(J = I e^x \mapsto x = {\textstyle {\frac{1}{2}}}\log (J^\top J)\) .
(iv) Let \((v,w)_A := \omega (Av,w)\)  denote the real Hilbert
structure on \(\mathcal {H}\)  defined by \(A\) . Then
\( \omega (x,y) = \omega (A(A^{-1}x), y) = (A^{-1}x,y)_A \) 
implies that \(A^{-1}\)  is skew-symmetric with respect to
\((\cdot ,\cdot )_A\) , and the same holds for \(A\)  itself.
Therefore \(-A^2 \ge 0\)  and
\(J := (-A^2)^{-1/2}A\)  is a complex structure leaving
\((\cdot ,\cdot )_A\)  invariant (cf. [6]}).
Then
\( \omega (Jv,v)= \omega ((-A^2)^{-1/2}Av,v)= ((-A^2)^{-1/2}v,v)_A \) 
is positive definite, so that \(J\)  is \(\omega \) -positive.
If \(J^{\prime }\)  is another \(\omega \) -positive complex structure commuting
with \(A\) , then the construction of \(J\)  shows that it also commutes
with \(J\) . Therefore \(JJ^{\prime }\)  is an involution and since
\(J\)  and \(J^{\prime }\)  are \(\omega \) -positive, the \(-1\) -eigenspace of this
involution is trivial, which leads to \(J^{\prime } = J\) .
(v) From the polar decomposition
\(\mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H}) = \mathop {\rm U{}}\nolimits (\mathcal {H}) \exp ({\mathfrak {p}}_2)\quad \mbox{ with } \quad {\mathfrak {p}}_2 := {\mathfrak {p}}\cap \mathfrak {sp}_{\rm res}(\mathcal {H})= \lbrace  x \in {\mathfrak {p}}\colon  \Vert x\Vert _2 < \infty \rbrace  \) 
we derive that
\({\mathcal {I}}_\omega ^{\rm res} = I e^{{\mathfrak {p}}_2}.\) 
For the entire function \(F(z) := \frac{e^z - 1}{z}\) , we then have
\( Ie^x - I = I F(x) x, \) 
Since \(F(x)\)  is invertible for the real symmetric operator
\(x\)  by the Spectral Mapping Theorem, it follows that
\(Ie^x - I\)  is Hilbert–Schmidt if and only if \(x\) 
has this property.
For \(J = I e^x\)  we also observe that
\([I,J] = -e^x - Ie^x I = e^{-x} - e^{x}= e^{-x}(\mathbf {1} - e^{2x})= - e^{-x} F(2x) 2 x, \) 
so that \([I,J]\)  is Hilbert–Schmidt if and only if \(x\)  is.
This leads to \({\mathcal {I}}_\omega ^{\rm res} = {\mathcal {I}}_\omega \cap \mathfrak {sp}_{\rm res}(\mathcal {H})\) .
\(\Box \)
The following theorem is a useful tool when dealing with invariant
cones in symplectic Lie algebras.
Theorem 6.7   For the canonical open invariant cones
in \(\mathfrak {sp}(\mathcal {H})\) , \(\mathfrak {sp}_{\rm res}(\mathcal {H})\)  and \(\mathfrak {hsp}(\mathcal {H})\)  we have the following
conjugacy results:
(i)
\(W_{\mathfrak {sp}(\mathcal {H})} = \mathop {{\rm Ad}}\nolimits (\mathop {{\rm Sp}}\nolimits (\mathcal {H})) C_{{\mathfrak {u}}(\mathcal {H})}\)  and
\(C_{{\mathfrak {u}}(\mathcal {H})} = W_{\mathfrak {sp}(\mathcal {H})} \cap {\mathfrak {u}}(\mathcal {H})\) .
(ii)
\(W_{\mathfrak {sp}_{\rm res}(\mathcal {H})} := W_{\mathfrak {sp}\mathcal {H})} \cap \mathfrak {sp}_{\rm res}(\mathcal {H}) = \mathop {{\rm Ad}}\nolimits (\mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})) C_{{\mathfrak {u}}(\mathcal {H})}\) .

(iii)
\(W_{\mathfrak {hsp}(\mathcal {H})} = \mathop {{\rm Ad}}\nolimits (\mathop {\rm HSp{}}\nolimits (\mathcal {H}))({\mathbb {R}}\times \lbrace 0\rbrace  \times C_{{\mathfrak {u}}(\mathcal {H})})\) 
for the corresponding Lie group \(\mathop {\rm HSp{}}\nolimits (\mathcal {H}) = \mathop {{\rm Heis}}\nolimits (\mathcal {H}) \rtimes \mathop {{\rm Sp}}\nolimits (\mathcal {H})\) .
(iv)
\(W_{\mathfrak {hsp}_{\rm res}(\mathcal {H})} :=W_{\mathfrak {hsp}(\mathcal {H})} \cap \mathfrak {hsp}_{\rm res}(\mathcal {H})= \mathop {{\rm Ad}}\nolimits (\mathop {\rm HSp{}}\nolimits _{\rm res}(\mathcal {H}))({\mathbb {R}}\times \lbrace 0\rbrace  \times C_{{\mathfrak {u}}(\mathcal {H})})\) 
for \(\mathop {\rm HSp{}}\nolimits _{\rm res}(\mathcal {H}) := \mathop {{\rm Heis}}\nolimits (\mathcal {H}) \rtimes \mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})\) .


Proof.   
(i) If \(A \in \mathfrak {sp}(\mathcal {H})\)  is complex linear, then
\(A^* =- A\) , so that \(\omega (Av,v) = \mathop {{\rm Im}}\nolimits \langle Av,v\rangle = \langle -iAv,v\rangle \) .
Therefore \(A \in W_{\mathfrak {sp}(\mathcal {H})}\)  is equivalent to \(iA <\!\!< 0\) , i.e.,
\(W_{\mathfrak {sp}(\mathcal {H})} \cap {\mathfrak {u}}(\mathcal {H}) = C_{{\mathfrak {u}}(\mathcal {H})}\) .
For \(A \in W_{\mathfrak {sp}(\mathcal {H})}\)  we find with
Lemma REF (iv) a \(J \in {\mathcal {I}}_\omega \)  commuting with \(A\) .
For any \(g \in \mathop {{\rm Sp}}\nolimits (\mathcal {H})\)  with \(J = \mathop {{\rm Ad}}\nolimits (g)I\) ,
whose existence follows from Lemma REF (iii),
we conclude that
\(\mathop {{\rm Ad}}\nolimits (g)^{-1}A\)  commutes with \(I\) , hence is contained in
\(W_{\mathfrak {sp}(\mathcal {H})} \cap {\mathfrak {u}}(\mathcal {H}) = C_{{\mathfrak {u}}(\mathcal {H})}\) . This proves (i).
(ii) In view of Lemma REF (v), it suffices to show that
for any \(A \in W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}\)  the corresponding \(\omega \) -positive
complex structure \(J = (-A^2)^{-1/2}A\)  from Lemma REF (iv)
is contained in
\({\mathcal {I}}_\omega ^{\rm res}\) , i.e., \([I,J]\)  is Hilbert–Schmidt.
Since \(A^2\)  commutes with \(I\) , we have
\([I,J] = (-A^2)^{-1/2}[I,A]\) , so that the invertibility of
\((-A^2)^{-1/2}\)  implies that \(A \in \mathfrak {sp}_{\rm res}(\mathcal {H})\)  is equivalent
to \(J \in \mathfrak {sp}_{\rm res}(\mathcal {H})\) .
(iii) As we have seen in the proof of Lemma REF  above,
for each element \((c,x,A) \in W_{\mathfrak {hsp}(\mathcal {H})}\) , the Hamiltonian function
\(f(v) = c + \omega (x,v) + H_A(v)\) 
has a unique minimum in \(- A^{-1}x\) . Since the adjoint action of the
Heisenberg group \(\mathop {{\rm Heis}}\nolimits (\mathcal {H}) \subseteq \mathop {\rm HSp{}}\nolimits (\mathcal {H})\)  corresponds to
a translation action on \(\mathcal {H}\) , each adjoint orbit \(\mathcal {O}_X\)  in
\(W_{\mathfrak {hsp}(V,\omega )}\)  contains an element \(Y\)  whose corresponding
Hamiltonian function is minimal in 0, so that
\(Y \in {\mathbb {R}}\times \lbrace 0\rbrace  \times \mathfrak {sp}(\mathcal {H})\) .
In view of (i), each orbit in \(W_{\mathfrak {hsp}(V,\omega )}\)  meets the subalgebra
\({\mathbb {R}}\times \lbrace 0\rbrace  \times {\mathfrak {u}}(\mathcal {H})\) , and this proves (iii).
(iv) follows by combining the argument under (iii) with the
proof of (ii).
\(\Box \)
Invariant Lorentzian cones
If \(({\mathfrak {g}},\beta )\)  is a Lorentzian Lie algebra,
i.e., \(\beta \)  is an invariant Lorentzian form (which is negative definite
on a closed hyperplane), then
each half of the open double cone \(\lbrace  x \in {\mathfrak {g}}\colon  \beta (x,x) > 0 \rbrace \) 
is an open invariant cone in \({\mathfrak {g}}\) .
Example 6.8  
A particularly important example is \({\mathfrak {g}}= \mathop {{\mathfrak {sl} }}\nolimits _2({\mathbb {R}})\)  with
\(\beta (x,y) = -\mathop {{\rm tr}}\nolimits (xy)\) . Indeed, the basis
\(h =\begin{pmatrix}1 & \phantom{-}0 \\0 & -1\end{pmatrix},\quad u =\begin{pmatrix}\phantom{-}0 & 1 \\-1 & 0\end{pmatrix},\quad t =\begin{pmatrix}0 & 1 \\1 & 0\end{pmatrix}\) 
is orthogonal with
\( \beta (x h + y u + z t, x h + y u + z t) = -2 x^2 + 2 y^2 - 2 z^2. \) 
For \({\mathfrak {g}}= \mathop {{\mathfrak {sl} }}\nolimits _2({\mathbb {R}})\) , the adjoint group
coincides with the identity component \(\mathop {{\rm SO}}\nolimits ({\mathfrak {g}},\beta )_0\cong \mathop {{\rm SO}}\nolimits _{1,2}({\mathbb {R}})_0\) , which implies that the adjoint and
coadjoint orbits are the connected components of the level surfaces of the
associated quadratic form, and the 0-level surface of isotropic
vectors decomposes into the 0-orbit and two isotropic orbits
lying in the boundary of the double cone.
This description of the adjoint orbits implies in
particular that there are precisely two non-trivial open invariant
cones, namely the connected components of the set
\(\lbrace  x \colon  \beta (x,x) > 0\rbrace \) .

Remark 6.9 
Other examples of Lorentzian Lie algebras
arise as double extensions
from Lie algebras \({\mathfrak {g}}^0\) , endowed with an invariant scalar product
\(\kappa ^0\) :
If \(D \in \mathop {{\rm der}}\nolimits ({\mathfrak {g}}^0,\kappa ^0)\)  is a continuous skew-symmetric derivation,
then
\({\mathfrak {g}}:= {\mathbb {R}}\times {\mathfrak {g}}^0 \times {\mathbb {R}}\)  is a Lie algebra with respect to the
bracket
\([(z,x,t), (z^{\prime },x^{\prime },t^{\prime })] = (\kappa ^0(Dx,x^{\prime }), tDx^{\prime } - t^{\prime }Dx + [x,x^{\prime }],0)\) 
and the continuous symmetric bilinear form
\( \kappa ((z,x,t), (z^{\prime },x^{\prime },t^{\prime })) := zt^{\prime } + z^{\prime }t + \kappa ^0(x,x^{\prime }), \) 
is easily seen to be invariant.
The pair \(({\mathfrak {g}},\kappa )\)  is called a double
extension of \(({\mathfrak {g}}^0,\kappa ^0)\)  (cf. [10]}).

Example 6.10   (a) If \({\mathfrak {k}}\)  is a compact Lie algebra and
\({\mathfrak {g}}^0 := C^\infty ({\mathbb {S}}^1,{\mathfrak {k}})\)  is the corresponding loop algebra,
then we identify its elements with \(2\pi \) -periodic functions on \({\mathbb {R}}\) .
With an invariant scalar product \(\kappa _{\mathfrak {k}}\)  on \({\mathfrak {k}}\) , we obtain
the invariant scalar product
\( \kappa ^0(\xi ,\eta ) := \int _0^{2\pi } \kappa _{\mathfrak {k}}(\xi (t), \eta (t))\, dt \) 
on \({\mathfrak {g}}^0\) , and the derivation \(D\xi := \xi ^{\prime }\)  is skew-symmetric.
The corresponding double extension produces the (unitary forms) of the
untwisted affine Kac–Moody Lie algebras.
(b) If \({\mathfrak {g}}^0 := {\mathfrak {u}}_2(\mathcal {H})\)  is the Lie algebra of skew-hermitian
Hilbert–Schmidt operators on the Hilbert space \(\mathcal {H}\)  and
\(A = - A^* \in {\mathfrak {u}}(\mathcal {H})\) , then
\(\kappa ^0(x,y) := \mathop {{\rm tr}}\nolimits (xy^*) = -\mathop {{\rm tr}}\nolimits (xy)\)  is an invariant scalar product on
\({\mathfrak {g}}^0\)  and one obtains a double extension for the derivation
\(D(x) := [A,x]\) . It is non-trivial if and only if
\(A \notin {\mathbb {R}}i \mathbf {1} + {\mathfrak {u}}_2(\mathcal {H})\)  (cf. [8]}).

Problem 6.11  Classify infinite dimensional Lorentzian Lie algebras
\({\mathfrak {g}}\)  which are complete in the sense that for \(x \in {\mathfrak {g}}\) 
with \(\beta (x,x) > 0\)  the orthogonal space is a Hilbert space with
respect to \(-\beta \) . The construction in Example REF (b)
produces interesting examples.
Finite dimensional indecomposable Lorentzian Lie algebras
have been classified by Hilgert and Hofmann in [12]}.
The simple result is that an indecomposable finite dimensional
Lorentzian Lie algebra is either \(\mathop {{\mathfrak {sl} }}\nolimits _2({\mathbb {R}})\) , endowed with the
negative of its Cartan–Killing form, or a double extension
of an abelian Lie algebra, defined by an invertible skew-symmetric
derivation \(D\)  ([13]}).


Invariant cones of vector fields
Example 6.12 

Let \({\mathfrak {g}}= \mathcal {V}({\mathbb {S}}^1) = C^\infty ({\mathbb {S}}^1) \partial _\theta \) 
be the Lie algebra of smooth vector fields
on the circle \({\mathbb {S}}^1 \cong {\mathbb {R}}/{\mathbb {Z}}\) , where \(\partial _\theta :=\frac{\partial }{\partial \theta }\)  denotes the generator of the right rotations. Then
\( W_{\mathcal {V}({\mathbb {S}}^1)} := \Big \lbrace  f \partial _\theta \colon  f > 0 \Big \rbrace  \) 
is an open invariant cone (cf. Section ).

Example 6.13 

The preceding example has a natural higher dimensional generalization.
Let \((M,g)\)  be a compact Lorentzian manifold possessing a timelike
vector field \(T\) , i.e., \(g_m(T(m), T(m)) > 0\)  for every \(m \in M\) .
Then
\( W := \lbrace  X \in \mathcal {V}(M) \colon  (\forall m \in M)\,g(X,X) > 0, g(X,T) > 0 \rbrace  \) 
is an open convex cone in the Fréchet space \(\mathcal {V}(M)\)  and its
intersection with the subalgebra \(\mathfrak {conf}(M,g)\)  of conformal vector fields
is an open invariant cone.
For \(M = {\mathbb {S}}^1\)  all vector fields are conformal and we thus obtain
Example REF .


Invariant cones and symmetric Hilbert domains
Example 6.14 
 (a)
We recall from Example REF (g) the concept
of a symmetric Hilbert domain and write
\(G := \mathop {{\rm Aut}}\nolimits (\mathcal {D})_0\)  for the identity component of its automorphism
group. We assume w.l.o.g. that \(\mathcal {D}\)  is the open unit ball of a
complex Banach space \(V\)  (cf. [14]}, [15]}).
Let \(K = G \cap \mathop {{\rm GL}}\nolimits (V)\)  be the subgroup of linear
automorphisms of \(\mathcal {D}\) , i.e., the group of complex linear isometries of
\(V\)  and let \({\mathfrak {k}}= \mathop {\bf L{}}\nolimits (K)\)  be its Lie algebra. Then
\( W_{\mathfrak {k}}:= \lbrace  x \in {\mathfrak {k}}\colon  \Vert e^{i x}\Vert  < 1 \rbrace  \) 
is a pointed open convex cone in the subalgebra \({\mathfrak {k}}\) , and
\( W := \mathop {{\rm Ad}}\nolimits (G)W_{\mathfrak {k}}\) 
is a pointed open invariant convex cone in \({\mathfrak {g}}\) 
(cf. [16]}, and [17]} for the finite
dimensional case).
By definition, this open cone has the interesting
property that every orbit in \(W\)  meets \({\mathfrak {k}}\) .
(b) Because they will also show up in the following, we take a closer
look at some characteristic examples. In Examples REF (g),
we have seen that the group
\(G = \mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H}_+, \mathcal {H}_-)\)  acts naturally on the
circular symmetric Hilbert
domain \(\mathcal {D}:= \lbrace  z \in B_2(\mathcal {H}_+, \mathcal {H}_-) \colon  \Vert z\Vert  < 1\rbrace \) .
Here the stabilizer of 0 is
\( K := \mathop {\rm U{}}\nolimits (\mathcal {H}_-) \times \mathop {\rm U{}}\nolimits (\mathcal {H}_+), \) 
which acts by \((a,d)z = azd^{-1}\) . From the Hilbert space
isomorphism \(B_2(\mathcal {H}_+, \mathcal {H}_-)\cong \mathcal {H}_- \widehat{\otimes }\mathcal {H}_+^*\) , one now derives with
\(\Vert e^{ix_-} \otimes e^{-ix_+}\Vert &= \Vert e^{ix_-}\Vert \Vert e^{-ix_+}\Vert = \sup {\rm Spec}(i x_-) + \sup {\rm Spec}(-i x_+)\\&= \sup {\rm Spec}(i x_-) - \inf {\rm Spec}(i x_+)\) 

that
\( W_{\mathfrak {k}}= \lbrace  (x_-, x_+) \in {\mathfrak {k}}\cong {\mathfrak {u}}(\mathcal {H}_-) \times {\mathfrak {u}}(\mathcal {H}_+) \colon \sup {\rm Spec}(i x_-) < \inf {\rm Spec}(i x_+) \rbrace , \) 
and \(W = \mathop {{\rm Ad}}\nolimits (G)W_{\mathfrak {k}}\)  is the corresponding open invariant cone in
\({\mathfrak {g}}= {\mathfrak {u}}(\mathcal {H}_+, \mathcal {H}_-)\) . Since the action of \(G\)  on \(\mathcal {D}\)  is not
faithful, this cone has a non-trivial edge \(H(W) = {\mathbb {R}}i\mathbf {1}= \mathop {\bf L{}}\nolimits ()\) .
(c) For \(\mathcal {H}_+= \mathcal {H}_- = \mathcal {H}\)  and the subgroup
\(G = \mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H}) \subseteq \mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H}_+, \mathcal {H}_-)\) ,
we have \(K \cong \mathop {\rm U{}}\nolimits (\mathcal {H})\) , corresponding to the pairs of the form
\((a,a^{-\top }) \in \mathop {\rm U{}}\nolimits (\mathcal {H}) \times \mathop {\rm U{}}\nolimits (\mathcal {H})\) , and, accordingly,
\( W_{\mathfrak {k}}= \lbrace  x \in {\mathfrak {u}}(\mathcal {H})\ i x <\!\!< 0 \rbrace  = C_{{\mathfrak {u}}(\mathcal {H})}\) 
(Examples REF (h) and REF ).
From Theorem REF (ii) it now follows that
\(W = \mathop {{\rm Ad}}\nolimits (G)W_{\mathfrak {k}}= W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}.\) 


A general lemma
The following lemma captures the spirit of some arguments
in the previous constructions of invariant convex cones
in Lie algebras in a quite natural way.
F.i., it applies to finite dimensional simple
algebras as well as \(\mathcal {V}({\mathbb {S}}^1)\) .
Lemma 6.15 
Suppose that the element \(d \in {\mathfrak {g}}\)  has the following
properties:
(a)
The interior \(W_{\rm min}\)  of the invariant convex cone
generated by \(\mathcal {O}_d = \mathop {{\rm Ad}}\nolimits (G)d\)  is non-empty and different from \({\mathfrak {g}}\) .
(b)
There exists a continuous linear projection
\(p \colon  {\mathfrak {g}}\rightarrow {\mathbb {R}}d\)  which preserves every open and closed convex subset.
(c)
There exists an element \(x \in {\mathfrak {g}}\)  for which
\(p(\mathcal {O}_x)\)  is unbounded.

Then each non-empty open invariant cone \(W\)  contains \(W_{\rm min}\)  or
\(-W_{\rm min}\) , and for \(\lambda \in {\mathfrak {g}}^{\prime }\)  the following are equivalent
(i)
\(\lambda \in {\mathfrak {g}}^{\prime }_{\rm seq}\) , i.e.,
\(\mathcal {O}_\lambda \)  is semi-equicontinuous.
(ii)
\(\mathcal {O}_\lambda (d)\)  is bounded from below or above.
(iii)
\(\lambda \in W_{\rm min}^\star \cup - W_{\rm min}^\star \) .


Proof.   
If \(W \subseteq {\mathfrak {g}}\)  is an open invariant cone,
then \(p(W) \subseteq {\mathbb {R}}d\)  contains an open half line, so that (b)
implies that \(W\)  contains either \(d\)  or \(-d\) . Accordingly, we
then have \(W_{\rm min} \subseteq W\)  or \(-W_{\rm min} \subseteq W\) .
(iii) \(\Rightarrow \)  (i): As \(W_{\rm min}\)  is open,
\(W_{\rm min}^\star \cup - W_{\rm min}^\star \)  consists
of semi-equicontinuous coadjoint orbits (Example REF (b)).
(i) \(\Rightarrow \)  (ii): Let \(\lambda \in {\mathfrak {g}}^{\prime }_{\rm seq}\) .
Since \(B(\mathcal {O}_\lambda )^0\)  is an open invariant cone, (a) implies that
it either contains \(d\)  or \(-d\) , which is (ii).
(ii) \(\Rightarrow \)  (iii): Assume w.l.o.g. that
\(\inf \mathcal {O}_\lambda (d) > - \infty \) . We claim that
\(\mathcal {O}_\lambda \subseteq W_{\rm min}^\star \) , which is equivalent to
\(\mathcal {O}_\lambda (d) \ge 0\) .
To this end, we consider \(\mu \in {\mathfrak {g}}^{\prime }\)  defined by \(p(x) = \mu (x)d\) .
Then \(-d \notin W_{\rm min}\)  and (b) imply that
\(\mu \in W_{\rm min}^\star \) , so that
\(\mu (\mathcal {O}_d) \ge 0\) . If \(\mu (\mathcal {O}_d)\)  is bounded from above,
then \(\pm d \in B(\mathcal {O}_\mu )\) , so that the invariance of the convex
cone \(B(\mathcal {O}_\mu )\)  leads to \(\pm W_{\rm min}\subseteq B(\mathcal {O}_\mu )\) ,
and hence to \({\mathfrak {g}}= B(\mathcal {O}_\mu )\) , contradicting (c). We conclude that
\(\mu (\mathcal {O}_d)\)  is unbounded.
Applying (b) to the closed convex subset \(C := \overline{\mathop {{\rm conv}}\nolimits }(\mathcal {O}_d)\) ,
it follows that \([1,\infty [ \cdot d \subseteq C\) , and hence that
\(d \in \lim (C)\)  (Lemma REF (iii)).
As \(\lim (C)\)  is an invariant cone, (a) entails
\(W_{\rm min} \subseteq \lim (C)\) . We finally conclude that
\(\lambda \in B(\mathcal {O}_d) = B(C) \subseteq \lim (C)^\star \subseteq W_{\rm min}^\star \)  (Lemma REF )(vi)).
\(\Box \)
The preceding lemma applies in particular to finite dimensional
simple non-compact Lie algebras ([17]}):
Proposition 6.16  
For a finite dimensional simple non-compact
Lie algebra \({\mathfrak {g}}\) , the following assertions hold:
(i)
Every non-empty invariant convex subset \(C\ne \lbrace 0\rbrace \) 
has interior points.
(ii)
If \({\mathfrak {g}}\)  contains a proper open invariant convex cone,
then there exist minimal and maximal open invariant cones
\(W_{\rm min} \subseteq W_{\rm max}\)  such that for any other open invariant
cone \(W\)  we either have
\(W_{\rm min} \subseteq W \subseteq W_{\rm max} \quad \mbox{ or } \quad W_{\rm min} \subseteq -W \subseteq W_{\rm max}. \) 
In this case the set of semi-equicontinuous coadjoint orbit in
\({\mathfrak {g}}^{\prime }\)  coincides with \(W_{\rm min}^\star \cup - W_{\rm min}^\star \) .


Proof.   
(i) If \(\emptyset \ne  C \ne  \lbrace 0\rbrace \)  is invariant and convex,
then \(\mathop {{\rm span}}\nolimits (C)\)  is a non-zero ideal of \({\mathfrak {g}}\) , hence equal to \({\mathfrak {g}}\) .
If the affine subspace generated by \(C\)  is proper, then its translation
space is a hyperplane ideal of \({\mathfrak {g}}\) , contradicting the simplicity of
\({\mathfrak {g}}\) . Therefore \(C\)  generates \({\mathfrak {g}}\)  as an affine space, hence
has interior points because \(\mathop {{\rm dim}}\nolimits {\mathfrak {g}}< \infty \)  and
\(C\)  contains a simplex of maximal dimension.
(ii) Let \({\mathfrak {g}}= {\mathfrak {k}}\oplus {\mathfrak {p}}\)  be a Cartan decomposition of
\({\mathfrak {g}}\)  and \({\mathfrak {z}}:= {\mathfrak {z}}({\mathfrak {k}})\)  be the center of \({\mathfrak {k}}\) .
Then the existence of invariant cones implies that
\({\mathfrak {z}}= {\mathfrak {z}}_{\mathfrak {g}}({\mathfrak {k}}) = {\mathbb {R}}d\)  is one-dimensional ([17]}).
Therefore we have a fixed point projection
\(p_{\mathfrak {z}}\colon  {\mathfrak {g}}\rightarrow {\mathfrak {z}}\)  with respect to the action of the
compact subgroup \(e^{\mathop {{\rm ad}}\nolimits {\mathfrak {k}}}\) , and this projection
preserves all open and closed invariant convex subsets
(Proposition REF (a)).
If \(p_{\mathfrak {z}}(\mathcal {O}_x)\)  is bounded for every \(x \in {\mathfrak {g}}\) , then
\(p_{\mathfrak {z}}\) , considered as a linear functional on \({\mathfrak {g}}\) ,
has a bounded orbit. As \({\mathfrak {g}}\)  is simple, this implies that all
coadjoint orbits are bounded, and this contradicts the non-compactness
of \({\mathfrak {g}}\) . Therefore (a)-(c) in Lemma REF  are satisfied,
which implies that the set of semi-equicontinuous coadjoint
orbits is \(W_{\rm min}^\star \cup - W_{\rm min}^\star \) .
\(\Box \)
Connections to \(C^*\) -algebras
In this section we discuss two aspects of semibounded representation
theory in the context of \(C^*\) -algebras. The first one concerns
the momentum sets of restrictions of
representations of a unital \(C^*\) -algebra \(\mathcal {A}\)  to its unitary group
\(\mathop {\rm U{}}\nolimits (\mathcal {A})\) , and the second one concerns covariant representations
for \(C^*\) -dynamical systems defined by a Banach–Lie group \(H\)  acting
on a \(C^*\) -algebra \(\mathcal {A}\) . Using the results from Appendix ,
it follows that covariant representations lead to
smooth unitary representations of the Lie group
\(\mathop {\rm U{}}\nolimits (\mathcal {A}^\infty ) \rtimes H\) , so that spectral conditions for covariant
representations can
be interpreted in terms of semibounded representations and the
theory of invariant cones in Lie algebras becomes available.
In this subsection \(\mathcal {A}\)  denotes a unital \(C^*\) -algebra and
\(\mathop {\rm U{}}\nolimits (\mathcal {A})\)  its unitary group, considered as a Banach–Lie group
(Example REF (a)). We identify the state space
\(S(\mathcal {A}) := \lbrace  \varphi \in \mathcal {A}^{\prime } \colon  \varphi (\mathbf {1}) = \Vert \varphi \Vert  = 1\rbrace \subseteq \lbrace  \varphi \in \mathcal {A}^{\prime } \colon  \varphi ({\mathfrak {u}}(\mathcal {A})) \subseteq i {\mathbb {R}}\rbrace  \) 

of \(\mathcal {A}\)  with a subset of \({\mathfrak {u}}(\mathcal {A})^{\prime }\)  by mapping
\(\varphi \in S(\mathcal {A})\)  to the real-valued functional \(-i \varphi \vert _{{\mathfrak {u}}(\mathcal {A})}\in {\mathfrak {u}}(\mathcal {A})^{\prime }\)  (cf. [1]} for more details).
Representations of the unitary group
If \((\pi , \mathcal {H})\)  is a unitary representation of \(\mathop {\rm U{}}\nolimits (\mathcal {A})\) 
obtained by restricting an algebra representation, then
its momentum set is simply given by
\( I_\pi = \lbrace  \varphi \in S(\mathcal {A}) \colon  \varphi (\ker \pi ) =\lbrace 0\rbrace \rbrace = S(\mathcal {A}) \cap (\ker \pi )^\bot \cong S(\pi (\mathcal {A})) \) 
(cf. [1]}). In particular, the momentum set
is completely determined by the kernel of the representation \(\pi \) , resp.,
the \(C^*\) -algebra \(\pi (\mathcal {A})\) . This is why representations with the
same kernel are called physically equivalent in
algebraic Quantum Field Theory ([22]}).
In [23]} one finds a characterization
of the separable \(C^*\) -algebras \(\mathcal {A}\)  of type \(I\)  as those for
which irreducible representations are determined by their
kernels, hence by their momentum sets.
More generally, postliminal \(C^*\) -algebras
have this property ([23]}).
We conclude in particular that the existence
of separable \(C^*\) -algebras \(\mathcal {A}\)  which are not of type \(I\)  implies the
existence of non-equivalent irreducible representations
\((\pi _1, \mathcal {H}_1)\)  and \((\pi _2, \mathcal {H}_2)\)  with
\(\ker \pi _1 = \ker \pi _2\) , and hence with
\(I_{\pi _1} = I_{\pi _2}\) . We thus observe:
Theorem 7.1  Bounded irreducible unitary representations
of Banach–Lie groups
are in general not determined up to equivalence by their
momentum sets.

Proposition 7.2  (Momentum sets of irreducible representations) 
For irreducible representations \((\pi , \mathcal {H})\)  of a \(C^*\) -algebra
\(\mathcal {A}\) , the following assertions hold:
(i)
\(\mathop {\rm U{}}\nolimits (\mathcal {A})\)  acts transitively on \({\mathbb {P}}(\mathcal {H})\) , so that
\(\mathcal {O}_\pi := \mathop {{\rm im}}\nolimits (\Phi _\pi )\)  is a single coadjoint orbit with
\(I_\pi = \overline{\mathop {{\rm conv}}\nolimits }(\mathcal {O}_\pi )\) .
(ii)
\(\mathcal {O}_\pi \)  consists of pure states, i.e.,
\(\mathcal {O}_\pi \subseteq \mathop {{\rm Ext}}\nolimits (S(\mathcal {A}))\) .
(iii)
Two irreducible representations
\(\pi _1\)  and \(\pi _2\)  are equivalent if and only if
\(\mathcal {O}_{\pi _1} = \mathcal {O}_{\pi _2}\) .


Proof.   
(i) [23]}
(ii) [23]}
(iii) (cf. [23]})
From the naturality of the momentum map it follows that
equivalent representations have the same
orbits. The converse follows from the fact that an irreducible
representation can be recovered from any of its pure states \(\varphi \)  by
the GNS construction, and states in the same \(\mathop {\rm U{}}\nolimits (\mathcal {A})\) -orbit
lead to equivalent representations.
\(\Box \)
In [28]} these results are generalized to irreducible
representations of \(\mathop {\rm U{}}\nolimits (\mathcal {A})\)  occurring in tensor products
of algebra representations and their duals.
Examples 7.3  
(a) If \(\pi \)  is the identical representation
of the \(C^*\) -algebra \(\mathcal {A}= B(\mathcal {H})\)  on \(\mathcal {H}\) , then the corresponding
momentum map is given by
\( \Phi _\pi ([v])(x) = \frac{1}{i} \frac{\langle xv, v\rangle }{\langle v,v\rangle }= - i \mathop {{\rm tr}}\nolimits (xP_v), \) 
where \(P_v(w) = \frac{\langle w,v\rangle }{\langle v,v\rangle } v\)  is the orthogonal
projection onto \([v]\) . Clearly, this representation is faithful
and irreducible, so that \(I_\pi = S(\mathcal {A})\) . On
the other hand \(\mathcal {O}_\pi = \mathop {{\rm im}}\nolimits (\Phi _\pi )\)  can be identified
with the set of rank-one projections, as elements of the
dual space \(\mathcal {A}^{\prime }\) . With the trace pairing, we can embed
the subspace \(\mathop {{\rm Herm}}\nolimits _1(\mathcal {H})\)  of hermitian trace class operators
in to the dual \(\mathcal {A}^{\prime }\) . Then
\( \mathop {{\rm Herm}}\nolimits _1(\mathcal {H}) \cap S(\mathcal {A}) = \lbrace  S \in \mathop {{\rm Herm}}\nolimits _1(\mathcal {H}) \colon S \ge 0, \mathop {{\rm tr}}\nolimits S = 1\rbrace , \) 
and the Spectral Theorem for compact hermitian operators
implies that
\( \mathop {{\rm Ext}}\nolimits (S(\mathcal {A})) \cap \mathop {{\rm Herm}}\nolimits _1(\mathcal {H}) = \Phi _\pi ({\mathbb {P}}(\mathcal {H})) \) 
is a single coadjoint orbit. However, since the
Calkin algebra \(\mathcal {A}/K(\mathcal {H})\)  is non-trivial if \(\mathop {{\rm dim}}\nolimits \mathcal {H}= \infty \) ,
\(\mathcal {A}\)  also has pure states
vanishing on the ideal \(K(\mathcal {H})\)  of compact operators and
\(\mathop {\rm U{}}\nolimits (\mathcal {H})\)  does not act transitively on \(\mathop {{\rm Ext}}\nolimits (S(\mathcal {A}))\) .
(b) For the natural representation of \(G = \mathop {\rm U{}}\nolimits (\mathcal {H})\)  on
the symmetric powers \(S^n(\mathcal {H})\) , the elements
\([v^n]\) , \(0\ne v \in \mathcal {H}\) , form a single \(G\) -orbit on
which the complex group \(\mathop {{\rm GL}}\nolimits (\mathcal {H}) = \mathcal {A}^\times \)  acts holomorphically
(Proposition REF (i)). Since
\(\Phi _{S^n(\pi )}([v^n]) = n \Phi _\pi ([v]) \quad \mbox{ for } \quad 0 \ne  v\in \mathcal {H}, \) 
it thus follows from Example REF  that
\(I_{S^n(\pi )} = n I_\pi .\) 
For a generalization to more general subrepresentations of
\(\mathcal {H}^{\otimes n}\)  we refer to [28]}.

\(C^*\) -dynamical systems
Definition 7.4  
Let \(G\)  be a topological group and \(\mathcal {A}\)  a \(C^*\) -algebra.
A \(C^*\) -dynamical system is a triple
\((\mathcal {A}, G, \alpha )\) , where \(\alpha \colon  G \rightarrow \mathop {{\rm Aut}}\nolimits (\mathcal {A}), g \mapsto \alpha _g\) ,
is a homomorphism defining a continuous action of \(G\)  on \(\mathcal {A}\) .

Theorem 7.5 
 ([30]})
If \(G\)  is a Banach–Lie group and
\((\mathcal {A}, G, \alpha )\)  a \(C^*\) -dynamical system, then
the space \(\mathcal {A}^\infty \)  of smooth vectors is a
Fréchet algebra with respect to the locally convex
topology defined by the seminorms
\( p_n(a) := \sup \lbrace  \Vert {\tt d}\alpha (x_1) \cdots {\tt d}\alpha (x_n)a\Vert  \colon x_i \in {\mathfrak {g}}, \Vert x_i\Vert \le 1\rbrace , \quad n \in {\mathbb {N}}_0, \) 
and the action of \(G\)  on \(\mathcal {A}^\infty \)  is smooth.
If, in addition, \(\mathcal {A}\)  is unital, then \(\mathcal {A}^\infty \)  is a continuous
inverse algebra.

Now let \((\mathcal {A},H,\alpha )\)  be a \(C^*\) -dynamical system,
where \(H\)  is a Banach-Lie group and \(\mathcal {A}\)  is a unital \(C^*\) -algebra.
Then \(\mathop {\rm U{}}\nolimits (\mathcal {A}^\infty )\) 
carries a natural Fréchet–Lie group structure
(cf. Example REF (a)), and we can form the
semidirect product Lie group \(G := \mathop {\rm U{}}\nolimits (\mathcal {A}^\infty ) \rtimes H\) .
For the proof of the following theorem, we record
a general observation on invariant cones.
Lemma 7.6  
If \(W \subseteq {\mathfrak {g}}\)  is an open invariant
convex cone, then we have for each element \(x \in {\mathfrak {g}}\)  satisfying
\((\mathop {{\rm ad}}\nolimits x)^2 =0\)  the relation \([x,{\mathfrak {g}}] \subseteq H(W)\) .
In particular, \(x\)  is central if \(W\)  is pointed.

Proof.   
Let \(w \in W\) .
Then, for each \(t \in {\mathbb {R}}\) , we have
\(w + {\mathbb {R}}[x,w] = e^{{\mathbb {R}}\mathop {{\rm ad}}\nolimits x}w \subseteq W,\) 
so that \([x,w] \in H(W)\)  (Lemma REF ).
Now \({\mathfrak {g}}= W - W\)  leads to \([x,{\mathfrak {g}}] \subseteq H(W)\) .
If, in addition, \(W\)  is pointed, this implies that
\([x,{\mathfrak {g}}] = \lbrace 0\rbrace \) , i.e., \(x \in {\mathfrak {z}}({\mathfrak {g}})\) .
\(\Box \)
Theorem 7.7  
Let \((\pi ,\rho , \mathcal {H})\)  be a covariant representation of
\((\mathcal {A},H,\alpha )\) , i.e., \(\pi \)  is a non-degenerate representation of
\(\mathcal {A}\)  on \(\mathcal {H}\)  and \(\rho \)  is a unitary representation of \(H\)  satisfying
\( \pi (\alpha _g A) = \rho (g)\pi (A) \rho (g)^{-1} \quad \mbox{ for }\quad A \in \mathcal {A}, g \in H. \) 
Then the following assertions hold:
(i)
The corresponding representation
\(\widehat{\pi }(a,h) := \pi (a) \rho (h)\)  of \(G = \mathop {\rm U{}}\nolimits (\mathcal {A}^\infty ) \rtimes H\) 
is smooth if and only if
\(\rho \)  has this property.
(ii)
\(W_{\widehat{\pi }} = {\mathfrak {u}}(\mathcal {A}^\infty ) \times W_\rho \)  and
\(\widehat{\pi }\)  is semibounded if and only if \(\rho \)  has this property
if and only if \(C_{\widehat{\pi }} \ne \emptyset \) .
(iii)
If \(\mathcal {A}\)  is commutative and
\(\widehat{\pi }\)  is semibounded, then the identity component \(H_0\)  of \(H\) 
acts trivially on \(\pi (\mathcal {A})\) .


Proof.   
(i) For every \(H\) -smooth vector \(v \in \mathcal {H}\) , the smoothness of the
map \(\mathop {\rm U{}}\nolimits (\mathcal {A}) \times H \rightarrow \mathcal {H}, (a,g) \mapsto \pi (a)\rho (g)v\) 
follows from the smoothness of the action of the Banach–Lie group
\(\mathop {\rm U{}}\nolimits (\mathcal {A})\)  on \(\mathcal {H}\) . Since the inclusion \(\mathop {\rm U{}}\nolimits (\mathcal {A}^\infty ) \rightarrow \mathop {\rm U{}}\nolimits (\mathcal {A})\) 
is smooth, it follows that every \(H\) -smooth vector is smooth for
\(G\) , so that the corresponding unitary representation
\(\widehat{\pi }\colon  G \rightarrow \mathop {\rm U{}}\nolimits (\mathcal {H}), (a,g) \mapsto \pi (a)\rho (g)\) 
of \(G\)  is smooth whenever \(\rho \)  has this property.
(ii) Clearly \(i\mathbf {1} = {\tt d}\pi (i\mathbf {1}) \in {\tt d}\widehat{\pi }({\mathfrak {g}})\) , so that
\(\widehat{\pi }\)  is semibounded if and only if
\(C_{\widehat{\pi }} \ne \emptyset \)  (Proposition REF ).
Since \(\mathcal {A}\)  acts by bounded operators, we have
\({\mathfrak {u}}(\mathcal {A}^\infty ) \subseteq H(W_{\widehat{\pi }})\) , and thus
\( W_{\widehat{\pi }}= {\mathfrak {u}}(\mathcal {A}^\infty ) \times (W_{\widehat{\pi }} \cap {\mathfrak {h}})= {\mathfrak {u}}(\mathcal {A}^\infty ) \times W_\rho . \) 
Therefore \(\widehat{\pi }\)  is semibounded if and only if \(\rho \)  is semibounded.
(iii) As \({\mathfrak {u}}(\mathcal {A}^\infty )\)  is an abelian
ideal of \({\mathfrak {g}}\) , Lemma REF  implies that
\([{\mathfrak {u}}(\mathcal {A}^\infty ),{\mathfrak {h}}] \subseteq H(C_\pi ) = \ker {\tt d}\widehat{\pi }\) , i.e.,
that \(H_0\)  acts trivially on the \(C^*\) -algebra \(\pi (\mathcal {A})\) .
\(\Box \)
Remark 7.8 
A closely related fact is well-known in the
context of \(C^*\) -dynamical systems with the group
\(H = {\mathbb {R}}^d\) .
To explain the connection, let \(C \subseteq {\mathfrak {h}}^{\prime }\)  be a closed convex cone.
Then we say that \((\pi , \rho , \mathcal {H})\)  satisfies the
\(C\) -spectrum condition if \(-i{\tt d}\rho (x) \ge 0\)  for
\(x \in C^\star \) , i.e., \(I_\rho \subseteq C\) .
If \(C^\star \)  has interior points, we have just seen that
the \(C\) -spectrum condition implies that \(\widehat{\pi }\)  is semibounded,
and if \(\pi \)  is faithful and \(\mathcal {A}\)  is commutative,
this can only happen if \(H\)  acts trivially on \(\mathcal {A}\) 
(cf. [31]}).
To obtain non-trivial situations, one has to consider non-commutative
algebras. Typical examples arise for any
semibounded representation \((\rho , \mathcal {H})\)  of \(H\) 
for \(\mathcal {A}:= K(\mathcal {H})\)  (compact operators on \(\mathcal {H}\) ) and
\(\alpha _g(A) := \rho (g)A\rho (g)^{-1}\) .

Example 7.9  
(a) If \((\rho , \mathcal {H})\)  is a smooth representation of \(G\) ,
then the corresponding action of \(G\)  on \(K(\mathcal {H})\)  defined by
\(\alpha _g(A) := \rho (g)A\rho (g)^{-1}\)  also has a dense space \(K(\mathcal {H})^\infty \) 
of smooth vectors because for every pair \((v,w)\)  of smooth vectors
the corresponding rank-one operator \(P_{v,w}\) , defined by
\(P_{v,w}(x) := \langle x,w \rangle v\)  satisfies
\( \alpha _g P_{v,w} = P_{\rho (g)v, \rho (g)w}, \) 
which easily implies that its orbit map is smooth.
(b) If \(\mathcal {H}\)  is a complex Hilbert space and
\(\mathcal {A}:= \mathop {{\rm CAR}}\nolimits (\mathcal {H})\) , then the canonical action of the orthogonal
group \(\mathop {\rm O{}}\nolimits (\mathcal {H})\)  (of the underlying real Hilbert space)
on \(\mathcal {A}\)  is continuous and the subalgebra
\(\mathcal {A}^\infty \)  of smooth vectors is dense because it contains
\(a(\mathcal {H})\)  and hence the \(*\) -subalgebra generated by this subset
(cf. Section ).
(c) To find a similar situation for the CCR is not so obvious because
the Weyl algebra \(\mathop {{\rm CCR}}\nolimits (\mathcal {H})\) , i.e., the \(C^*\) -algebra
defined by the generators
\(W(f)\) , \(f \in \mathcal {H}\) , and the Weyl relations
\( W(f)^* = W(-f), \quad W(f) W(h) = e^{\frac{i}{2}\mathop {{\rm Im}}\nolimits \langle f, h \rangle } W(f+h) \) 
is a very singular object. The map \(W \colon  \mathcal {H}\rightarrow \mathop {{\rm CCR}}\nolimits (\mathcal {H})\) 
is discontinuous, even on every ray in \(\mathcal {H}\) 
([32]}).
Since the action of the symplectic
group \(\mathop {{\rm Sp}}\nolimits (\mathcal {H})\)  preserves these relations, it acts by
\(\alpha _g(W(f)) = W(gf)\)  on \(\mathop {{\rm CCR}}\nolimits (\mathcal {H})\) , but this action
is highly discontinuous (cf. Section ).
It seems that one possible way out of this dilemma is to find
suitable \(C^*\) -algebras consisting of operators with a more regular
behavior than \(\mathop {{\rm CCR}}\nolimits (\mathcal {H})\) . For interesting recent results
in this direction we refer to Georgescu's work
[33]}. Another step in this direction is the construction
of a \(C^*\) -algebra \(\mathcal {A}\)  for each countably dimensional symplectic space
whose representations correspond to those representations of the
corresponding Weyl relations which are continuous on each one-parameter
group ([34]}).

The Virasoro algebra and vector fields on \({\mathbb {S}}^1\)
In this section we discuss invariant cones
in the Lie algebra \(\mathcal {V}({\mathbb {S}}^1)\)  of smooth vector fields on the circle
and its (up to isomorphy unique) non-trivial central extension
\(\mathop {{\mathfrak {vir}}}\nolimits \) , the Virasoro algebra. For \(\mathcal {V}({\mathbb {S}}^1)\)  we show that, up to sign,
there is only one open invariant convex cone given by vector fields
of the form \(f \frac{\partial }{\partial \theta }\)  with \(f > 0\) .
As is well-known on the Lie algebra level, all unitary highest weight
representations of \(\mathcal {V}({\mathbb {S}}^1)\) , resp., the subalgebra of vector fields
for which \(f\)  is a finite Fourier polynomial, are trivial.
On the group level we show the closely related result that
all semibounded unitary representations of \(\mathop {{\rm Diff}}\nolimits ({\mathbb {S}}^1)_+\)  are trivial.
This is the main reason for the Virasoro algebra and the corresponding
simply connected group \(\mathop {{\rm Vir}}\nolimits \)  playing a more important role
in mathematical physics than \(\mathop {{\rm Diff}}\nolimits ({\mathbb {S}}^1)_+\)  itself
(cf. [35]}, [36]}, [37]}).
For \(\mathop {{\mathfrak {vir}}}\nolimits \)  we prove a convexity theorem for
adjoint and coadjoint orbits which provides complete
information on invariant cones and permits us to
determine the momentum sets of the unitary highest weight representations
of \(\mathop {{\rm Vir}}\nolimits \)  and to show that they are semibounded.
The invariant cones in \(\mathcal {V}({\mathbb {S}}^1)\)
We consider \({\mathbb {S}}^1\)  as the quotient \({\mathbb {R}}/2\pi {\mathbb {Z}}\)  and identify
smooth functions on \({\mathbb {S}}^1\)  with the corresponding
\(2\pi \) -periodic smooth functions on \({\mathbb {R}}\) , where the coordinate is denoted
by \(\theta \) . Accordingly \(\partial _\theta := \frac{d}{d\theta }\) 
is the vector field generating the rigid rotations of \({\mathbb {S}}^1\) .
We write \(G := \mathop {{\rm Diff}}\nolimits ({\mathbb {S}}^1)_+^{\rm op}\) 
for the group of orientation preserving
diffeomorphisms endowed with the group structure defined by
\(\varphi \cdot \psi := \psi \circ \varphi \) , so that
\({\mathfrak {g}}= \mathcal {V}({\mathbb {S}}^1) = C^\infty ({\mathbb {S}}^1) \partial _\theta \) 
is its Lie algebra (cf. Examples REF (e)).
We represent orientation preserving
diffeomorphisms of \({\mathbb {S}}^1\)  by smooth functions
\(\varphi \colon  {\mathbb {R}}\rightarrow {\mathbb {R}}\)  satisfying
\(\varphi (\theta +2\pi ) = \varphi (\theta ) + 2\pi \)  for \(\theta \in {\mathbb {R}}\)  and \(\varphi ^{\prime } > 0\) .
In the following it will be convenient to consider
the spaces
\(\mathcal {F}_s({\mathbb {S}}^1) := C^\infty ({\mathbb {S}}^1) (d\theta )^s, \quad s \in {\mathbb {R}}, \) 
of \(s\) -densities on \({\mathbb {S}}^1\) . Here \((d\theta )^s\)  denotes the canonical
section of the \(s\) -density bundle of \({\mathbb {S}}^1\)  and \(G\)  acts on
\(\mathcal {F}_s({\mathbb {S}}^1)\)  by pullbacks
\(\varphi ^*(u (d\theta )^s)= (u \circ \varphi ) (\varphi ^*d\theta )^s= (u \circ \varphi ) (\varphi ^{\prime })^s (d\theta )^s.\) 
The corresponding derived action of the Lie algebra \(\mathcal {V}({\mathbb {S}}^1)\) 
is given by the Lie derivative
\(\mathcal {L}_{f\partial _\theta }(u(d\theta )^s)= (fu^{\prime } + s f^{\prime }u)(d\theta )^s.\) 
The space \(\mathcal {F}_1\)  is the space of 1-forms and \(\mathcal {F}_{-1} \cong \mathcal {V}({\mathbb {S}}^1)\)  is the space of vector fields on which
(REF ) describes the adjoint action. We have
equivariant multiplication maps \(\mathcal {F}_s \times \mathcal {F}_t \rightarrow \mathcal {F}_{s+t}\) ,
and an invariant integration map
\( I \colon  \mathcal {F}_1 \rightarrow {\mathbb {R}}, \quad f d\theta \mapsto \int _0^{2\pi } f(\theta )\, d\theta ,\) 
which leads to an invariant pairing
\(\mathcal {F}_{-1} \times \mathcal {F}_2 \rightarrow {\mathbb {R}},\)  and hence to an equivariant embedding
\( \mathcal {F}_2 \hookrightarrow \mathcal {V}({\mathbb {S}}^1)^{\prime } = {\mathfrak {g}}^{\prime }. \) 
Its image is called the smooth dual of \({\mathfrak {g}}\). Identifying it
with \(\mathcal {F}_2\) , the coadjoint action of \(G\)  on \({\mathfrak {g}}^{\prime }\) 
corresponds to the natural action on \(\mathcal {F}_2\) .
In view of (REF ), the adjoint action
clearly preserves the open cone
\( W_{\mathcal {V}({\mathbb {S}}^1)} := \lbrace  f \partial _\theta \colon  f > 0, f \in C^\infty ({\mathbb {S}}^1) \rbrace  \) 
of all vector fields corresponding to positive functions.
Since every positive function \(f\)  with \(\int _0^{2\pi } f\, d\theta = 2\pi \) 
arises as \(\varphi ^{\prime }\)  for some \(\varphi \in G\) , each \(G\) -orbit
in \(W_{\mathcal {V}({\mathbb {S}}^1)}\)  intersects the (maximal) abelian subalgebra
\({\mathfrak {t}}:= {\mathbb {R}}\partial _\theta \) . We also have a projection map
\( p_{\mathfrak {t}}\colon  {\mathfrak {g}}\rightarrow {\mathfrak {t}}, \quad f \partial _\theta \mapsto \frac{1}{2\pi }\int _0^{2\pi } f(s)\, ds \cdot \partial _\theta = \int _T \mathop {{\rm Ad}}\nolimits (\varphi )(f\partial _\theta )\, d\mu _T(\varphi ),\) 
where \(T := \exp ({\mathfrak {t}})\cong  is the group of rigid rotationsand \) T\( the normalized Haar measure on \) T\(.\) The following lemma provides some fine information on the convex geometry
of adjoint orbits in \(W_{\mathcal {V}({\mathbb {S}}^1)}\) .
Lemma 8.1   The function
\(\chi \colon  W_{\mathcal {V}({\mathbb {S}}^1)} \rightarrow {\mathbb {R}}, f \partial _\theta \mapsto \chi (f) := \frac{1}{2\pi }\int _0^{2\pi } \frac{1}{f}\, d\theta \) 
has the following properties:
(i)
It is smooth, \(G\) -invariant and strictly convex.
(ii)
If the sequence \((f_n)\)  in \(W_{\mathcal {V}({\mathbb {S}}^1)}\)  converges to
\(f \in \partial W_{\mathcal {V}({\mathbb {S}}^1)}\) , then \(\chi (f_n) \rightarrow \infty \) .
(iii)
\(\chi (f + g) \le \chi (f)\)  for \(f,g \in W_{\mathcal {V}({\mathbb {S}}^1)}\) .
(iv)
For each \(c > 0\) , the set
\(I_c := \lbrace  x \in W_{\mathcal {V}({\mathbb {S}}^1)} \colon  \chi (x) \le \frac{1}{c}\rbrace \) 
is an invariant closed convex subset of \({\mathfrak {g}}\)  with
\(\lim (I_c) = \overline{W_{\mathcal {V}({\mathbb {S}}^1)}}\) . Its boundary is a single
orbit
\( \mathcal {O}_{c\partial _\theta } = \partial I_c = \mathop {{\rm Ext}}\nolimits (I_c)= \Big \lbrace  f \in W_{\mathcal {V}({\mathbb {S}}^1)} \colon  \chi (f) = \frac{1}{c}\Big \rbrace \) 
which coincides with its set of extreme points and satisfies
\(I_c = \overline{\mathop {{\rm conv}}\nolimits }(\mathcal {O}_{c\partial _\theta })\) .
(v)
\(p_{\mathfrak {t}}(\mathcal {O}_{c\partial _\theta })= p_{{\mathfrak {t}}}(I_c) = [c,\infty [ \cdot \partial _\theta \)  and
the only inverse image of the “minimal value” \(c\partial _\theta \)  is the
element \(c\partial _\theta \)  itself.


Proof.   
(i) The \(G\) -invariance of \(\chi \)  follows immediately
from the Substitution Rule.
The function \(\chi \)  is convex because inversion is a convex function on
\({\mathbb {R}}_+^\times \)  and integrals of convex functions are convex.
It is smooth because on \(W_{\mathcal {V}({\mathbb {S}}^1)}\)  pointwise inversion is a smooth
operation (since \(C^\infty ({\mathbb {S}}^1,{\mathbb {R}})\)  is a real continuous inverse
algebra; [38]}), and integration
is a continuous linear functional.
To verify that \(\chi \)  is strictly convex, we observe that
\((\partial _h \chi )(f) = -\frac{1}{2\pi }\int _0^{2\pi } \frac{h}{f^2}\, d\theta \quad \mbox{ and } \quad (\partial _h^2 \chi )(f) = \frac{1}{\pi }\int _0^{2\pi } \frac{h^2}{f^3}\, d\theta ,\) 
which is positive definite for each \(f > 0\) .
(ii) Now we turn to the boundary behavior of \(\chi \) . Suppose that
the sequence \((f_n)\)  in \(W_{\mathcal {V}({\mathbb {S}}^1)}\)  tends to a boundary point \(f \in \partial W_{\mathcal {V}({\mathbb {S}}^1)}\) .
Then \(f(\theta _0) = 0\)  for some \(\theta _0 \in [0,2\pi [\) , and
\(f \ge 0\)  implies that we also have \(f^{\prime }(\theta _0) = 0\) , hence
\(f(\theta ) \le C (\theta -\theta _0)^2\)  in a compact \(\delta \) -neighborhood \(U\)  of \(\theta _0\) .
Given \(\varepsilon > 0\) , we eventually have
\(f_n \le 2C(\theta -\theta _0)^2 + \varepsilon \)  on \(U\)  (here we use \(C^2\) -convergence),
and therefore
\(2\pi \chi (f_n)\ge \int _{\theta _0-\delta }^{\theta _0+\delta } \frac{1}{f_n(\theta )}\, d\theta \ge \int _{-\delta }^{\delta } \frac{1}{2C \theta ^2 + \varepsilon }\, d\theta . \) 
Since
\(\int _{-\delta }^{\delta } \frac{1}{\theta ^2}\, d\theta = \infty ,\) 
the Monotone Convergence Theorem implies that
\( \lim _{\varepsilon \rightarrow 0} \int _{-\delta }^{\delta } \frac{1}{2C\theta ^2+\varepsilon }\, d\theta = \infty , \) 
and therefore that \(\chi (f_n)\rightarrow \infty \) . It also follows that
\(\int _0^{2\pi } \frac{1}{f(\theta )}\, d\theta = \infty \) .
(iii) If \(f,g \in W_{\mathcal {V}({\mathbb {S}}^1)}\) , then \(\frac{1}{f+g} \le \frac{1}{f}\) 
implies the assertion.
(iv) From (ii) we derive that \(I_c\)  is closed in \({\mathfrak {g}}\) .
Its invariance follows from the invariance of \(\chi \)  and
its convexity from the convexity of \(\chi \) .
The boundary of \(I_c\)  is a level set of \(\chi \) , and
since every orbit in \(W_{\mathcal {V}({\mathbb {S}}^1)}\)  meets \({\mathbb {R}}_+ \partial _\theta \)  in a unique point
and \(\chi \vert _{{\mathbb {R}}_+^\times \partial _\theta }\)  is injective, it follows that
\(\partial I_c = \mathcal {O}_{c \partial _\theta }\) .
The fact that \(\chi \)  is strictly convex further implies that
\(\partial I_c \subseteq \mathop {{\rm Ext}}\nolimits (I_c)\) , and since the converse inclusion is trivial,
equality follows.
From (iii) we derive that \(W_{\mathcal {V}({\mathbb {S}}^1)} \subseteq \lim (I_c)\) ,
so that the closedness of \(\lim (I_c)\) 
(Lemma REF (i)) implies that
\(\overline{W_{\mathcal {V}({\mathbb {S}}^1)}} \subseteq \lim (I_c)\) . Now equality follows from
\(\lim (I_c) \subseteq \lim (W_{\mathcal {V}({\mathbb {S}}^1)}) = \overline{W_{\mathcal {V}({\mathbb {S}}^1)}}\) 
(Lemma REF (ii)).
To see that \(I_c\)  coincides with the closed convex hull
\(D := \overline{\mathop {{\rm conv}}\nolimits }(\mathcal {O}_{c \partial _\theta })\) ,
we first observe that we trivially
have \(D \subseteq I_c\) . Next we note that
\(\partial _\theta \)  is contained in the 3-dimensional subalgebra
\({\mathfrak {s}}:= \mathop {{\rm span}}\nolimits \lbrace  1, \cos (\theta ), \sin (\theta ) \rbrace  \partial _\theta \cong \mathop {{\mathfrak {sl} }}\nolimits _2({\mathbb {R}}) \) 
corresponding to the action of \(\mathop {{\rm SL}}\nolimits _2({\mathbb {R}})\)  on
\({\mathbb {P}}_1({\mathbb {R}}) \cong {\mathbb {S}}^1\) . In \({\mathfrak {s}}\) 
the element \(\partial _\theta \)  corresponds to the matrix
\(u =\begin{pmatrix}\phantom{-}0 & 1 \\-1 & 0\end{pmatrix}, \) 
so that Example REF  implies that the corresponding
group \(S := \langle \exp {\mathfrak {s}}\rangle \cong \mathop {{\rm PSL}}\nolimits _2({\mathbb {R}})\)  satisfies
\(p_{\mathfrak {t}}(\mathop {{\rm Ad}}\nolimits (S)\partial _\theta ) = [1,\infty [ \cdot \partial _\theta \) 
(cf. Example REF ) and from that we derive in
particular that
\([c,\infty [ \cdot \partial _\theta \subseteq p_{\mathfrak {t}}(\mathcal {O}_{c \partial _\theta })\subseteq D,\) 
so that \(I_c = \mathop {{\rm Ad}}\nolimits (G)([c,\infty [\cdot \partial _\theta )\)  leads to
\(I_c \subseteq D\) .
(v) We have already seen in (REF ) above that
\( (\partial _h \chi )(c) = - \frac{1}{2\pi c^2} \int _0^{2\pi } h\, d\theta \) , so that
\(p_{\mathfrak {t}}^{-1}(c\partial _\theta )\)  is a tangent hyperplane of the strictly convex
set \(I_c\) . This implies that \(c \partial _\theta \)  is the unique minimum of the
linear functional \(-{\tt d}\chi (c)\)  on \(I_c\)  and hence that
\(p_{{\mathfrak {t}}}^{-1}(c\partial _\theta ) \cap I_c = \lbrace c\partial _\theta \rbrace \) .
We also conclude that
\(p_{\mathfrak {t}}(I_c) \subseteq [c,\infty [ \cdot \partial _\theta \) ,
so that (v) follows from (REF ).
\(\Box \)
Remark 8.2  
The topological dual \(\mathcal {V}({\mathbb {S}}^1)^{\prime }\)  of \(\mathcal {V}({\mathbb {S}}^1) = C^\infty ({\mathbb {S}}^1)\partial _\theta \)  naturally identifies with the space
of distributions on \({\mathbb {S}}^1\) .
Then each \(\lambda \in W_{\mathcal {V}({\mathbb {S}}^1)}^\star \)  is a distribution satisfying
\(\lambda (f) \ge 0\)  for \(f \ge 0,\) 
and this implies that \(\lambda \)  extends continuously from
\(C^\infty ({\mathbb {S}}^1)\)  to the Banach space \(C({\mathbb {S}}^1)\)  and thus defines a
(finite) positive Radon measure
on \({\mathbb {S}}^1\)  (cf. [39]}).
This shows that the functionals in \(W_{\mathcal {V}({\mathbb {S}}^1)}^\star \)  satisfy a strong
regularity condition.

Theorem 8.3 (Classification of open invariant cones in
\(\mathcal {V}({\mathbb {S}}^1)\) ) 
The two open cones \(\pm W_{\mathcal {V}({\mathbb {S}}^1)}\)  are the only
non-empty proper open invariant cones in \(\mathcal {V}({\mathbb {S}}^1)\) .

Proof.   
Let
\(C \subseteq \mathcal {V}({\mathbb {S}}^1)\)  be a non-emptry open invariant cone. Then \(C\) 
is in particular invariant under the adjoint action
of the rotation group
\(T \cong , generated by the vector field\) d := \(. Averaging over \) T\(, we see thateither \) d\( or \) -d\( is contained in \) C\((Proposition~\ref {prop:project}), which leads to\) WV(S1) C\( or \) -WV(S1) C\(.\) Let us assume that \(W_{\mathcal {V}({\mathbb {S}}^1)} \subseteq C\) .
Now we apply the same argument to the dual cone \(C^\star \subseteq W_{\mathcal {V}({\mathbb {S}}^1)}^\star \)  (Proposition REF (b)).
If \(C\)  is proper, then \(C^\star \)  contains a non-zero functional
\(\lambda \) , and then \(d \in C\)  leads to \(\lambda (d) > 0\) .
Averaging over \(T\)  now leads to a \(T\) -invariant
functional \(\lambda ^* \in C^\star \subseteq W_{\mathcal {V}({\mathbb {S}}^1)}^\star \)  satisfying
\(\lambda ^*(d) = \lambda (d) > 0\) . We conclude that \(\lambda ^*\)  is a positive
multiple of the invariant measure \(\mu = d \theta \) 
on
\({\mathbb {S}}^1 \cong {\mathbb {R}}/2\pi {\mathbb {Z}}\)  (Remark REF ),
which corresponds to the constant function \(1 \in \mathcal {F}_2\) ,
so that
\( \mathop {{\rm Ad}}\nolimits ^*(\varphi ) \mu = (\varphi ^{\prime })^2 \mu \) 
follows from (REF ).
For any \(f \partial _\theta \in C\)  we now find
\(\int _0^{2\pi } \psi (\theta )^2 f(\theta )\, d\theta \ge 0\) 
for each positive function \(\psi \)  with \(\int _0^{2\pi } \psi (\theta )\, d\theta =2\pi \)  (these are precisely the functions occurring as \(\varphi ^{\prime }\)  for an
orientation preserving diffeomorphism), and this implies the
corresponding relation for all non-negative functions
\(\psi \)  with the integral \(2\pi \) .
If \(f(\theta _0) < 0\) , then we may choose \(\psi \)  supported by
the set \(\lbrace  f < 0\rbrace \)  and obtain a contradiction.
This proves that \(C \subseteq \overline{W_{\mathcal {V}({\mathbb {S}}^1)}}\)  and hence that \(C = W_{\mathcal {V}({\mathbb {S}}^1)}\) 
because \(W_{\mathcal {V}({\mathbb {S}}^1)} = (\overline{W_{\mathcal {V}({\mathbb {S}}^1)}})^0\) 
(Lemma REF ).
\(\Box \)
Proposition 8.4  
For \(\lambda \in \mathcal {V}({\mathbb {S}}^1)^{\prime }\)  the following are equivalent
(i)
\(\mathcal {O}_\lambda \)  is semi-equicontinuous.
(ii)
\(\inf \mathcal {O}_\lambda (\partial _\theta ) > -\infty \)  or
\(\sup \mathcal {O}_\lambda (\partial _\theta ) < \infty \) .
(iii)
\(\lambda \in W_{\mathcal {V}({\mathbb {S}}^1)}^\star \cup - W_{\mathcal {V}({\mathbb {S}}^1)}^\star \) .

In particular, \(\inf \mathcal {O}_\lambda (\partial _\theta ) > -\infty \)  implies
\(\mathcal {O}_\lambda (\partial _\theta ) \ge 0\) .

Proof.   
We have to verify the assumptions of Lemma REF 
for \(d := \partial _\theta \) .
First, \(W_{\mathcal {V}({\mathbb {S}}^1)}\)  is the open convex cone generated by
the orbit \(\mathcal {O}_d\) . Second, the projection
\(p_{\mathfrak {t}}\colon  {\mathfrak {g}}\rightarrow {\mathbb {R}}d = {\mathfrak {t}}\)  is the fixed point projection for the
adjoint action of the circle group \(T\) , hence preserves
open and closed convex subsets (Proposition REF ).
Finally, we recall from Lemma REF (v)
that \(p_{\mathfrak {t}}(\mathcal {O}_d) = [1,\infty [d\)  is unbounded. Now the assertion
follows from Lemma REF .
\(\Box \)
Theorem 8.5  
The quotient \(M := G/T\)  of the group
\(G = \mathop {{\rm Diff}}\nolimits ({\mathbb {S}}^1)_+\)  by the subgroup \(T\)  of rigid rotations
carries the structure of a complex Fréchet manifold on which
\(G\)  acts smoothly by holomorphic maps.
Here the tangent space \(T_{x_0}(G/T)\)  in the base point
\(x_0 = \mathbf {1} T\)  is canonically identified with
\({\mathfrak {g}}_{\mathfrak {p}}\) , where
\( {\mathfrak {p}}= \Big \lbrace  f \partial _\theta \in C^\infty ({\mathbb {S}}^1, \partial _\theta \colon f = \sum _{n \le 0}^\infty a_n e^{in\theta }, a_n\in \rbrace . \) 
For each element \(x \in W_{\mathcal {V}({\mathbb {S}}^1)}\) , the flow on \(M\)  defined by
\((t,gT) \mapsto \exp (tx)g T\)  extends to a smooth flow on the upper
half plane \(+\)  which is holomorphic on
\(+^0 \times M\) .

Proof.   
The complex structure on \(M\)  has been discovered
by Kirillov and Yuriev (cf. [40]}, [41]}, [42]};
see in particular [43]} for rigorous arguments concerning the
complex Fréchet manifold structure).
Complex structures always
come in pairs, and we therefore write
\(\overline{M}\)  for the same smooth manifold, endowed with the
opposite complex structure.
According to [41]}, the complex manifold structure on \(\overline{M}\) 
can be obtained by identifying it with the space
\(\mathcal {F}_{\rm reg}\)  of normalized regular univalent functions
\(f \colon  \mathcal {D}\rightarrow , where \) D\( is the open unit disc in \) .
These are the injective holomorphic maps \(f \colon  \mathcal {D}\rightarrow extending smoothly to theclosure of \) D\( and satisfying \) f(0) = 0\( and \) f'(0) = 1\((cf.\ \cite [Sect.~6.5.6]{GR07} for a detailed discussion).Here the complex structure is determined by its action on\) Tx0(G/T) TidD(Freg) g/t\(:\)\( I[(e^{in\theta }+ e^{-in\theta })\partial _\theta ]= [(-i e^{in\theta }+ i e^{-in\theta })\partial _\theta ]\quad \mbox{ for } \quad n > 0. \)\(\) Since each \(x \in W_{\mathcal {V}({\mathbb {S}}^1)}\)  is conjugate to a multiple of \(\partial _\theta \) ,
it suffices to assume that \(x = \partial _\theta \)  is the generator
of the rigid rotations of \({\mathbb {S}}^1\) . In this case the action
of the one-parameter group \(T = \exp _G({\mathbb {R}}x)\)  on \(\mathcal {F}_{\rm reg}\) 
is given explicitly by
\( R_\alpha f = R_\alpha \circ f \circ R_{-\alpha }, \quad \mbox{where}\quad R_\alpha z = e^{i\alpha }z, \alpha \in {\mathbb {R}}\) 
(cf. [45]}).
If \(\mathop {{\rm Im}}\nolimits \alpha \le 0\) , then \(|e^{-i\alpha }| \le 1\) , so that
\(R_\alpha (\mathcal {D}) \subseteq \mathcal {D}\)  implies that
\(R_\alpha f\)  can still be defined as above, is continuous on
\(- := -+\)  and depends holomorphically on \(\alpha \)  for \(\mathop {{\rm Im}}\nolimits \alpha < 0\) .
This implies the holomorphic extension to \(-\)  for the
complex manifold \(\overline{M}\) . For the manifold \(M\)  we therefore
obtain an extension to \(+\) .
\(\Box \)
The holomorphic extension of actions of one-parameter groups
on \(G/T\)  can be carried much further. As shown by Neretin in [46]},
one even has a “complex semigroup” containing \(G\)  in its boundary
which acts on \(G/T\) .
Below we shall use the preceding theorem to identify the momentum
sets for the unitary highest weight representations of \(\mathop {{\mathfrak {vir}}}\nolimits _.\)
Definition 8.6  For the following we recall some algebraic
aspects of \(\mathcal {V}({\mathbb {S}}^1)\) .
In the complexification
\(\mathcal {V}({\mathbb {S}}^1)_, we consider the elements\begin{equation}d_n := i e^{i n\theta } \partial _\theta , \quad n \in {\mathbb {Z}},\end{equation}satisfying the commutation relations\)\( [d_n, d_m] = (n-m) d_{n+m}. \)\(The standard involution on this Lie algebrais given by \) (f )* = -f \(,so that \) x* = - x\( describes the elements of \) V(S1)\(.Note that \) dn* = d-n\( and in particular\) d0* = d0\(, so that \) d0 = i \(is a hermitian element (cf.\ \cite [p.~9]{KR87}).\)
Theorem 8.7  
All semibounded unitary representations
of the group \(\mathop {{\rm Diff}}\nolimits ({\mathbb {S}}^1)_+\)  are trivial.

Proof.   
Let \((\pi , \mathcal {H})\)  be a semibounded unitary representation
of \(G = \mathop {{\rm Diff}}\nolimits ({\mathbb {S}}^1)_+^{\rm op}\) . Then \(W_\pi \ne \emptyset \) , and
in view of Theorem REF , we may w.l.o.g. assume that \(- \partial _\theta \in -W_{\mathcal {V}({\mathbb {S}}^1)} \subseteq W_\pi \) ,
so that the spectrum of the image of
\(d_0 := i \partial _\theta \in {\mathfrak {g}}_ under the derived representationis bounded from below. In view of \) G(2i d0) = 1\(, it is containedin \) Z\( and Proposition~\ref {prop:maxsmoothvect} implies theexistence of a smooth unit vector \) v H\( which is aneigenvector for the minimal eigenvalue \) h\( of \) d0\(.\) Now the relation \([d_0, d_n] = - n d_n\)  implies
that \(d_n.v = 0\)  for each \(n > 0\) .
For \(n > 0\)  we then obtain
\(\langle d_{-n}v, d_{-n}v\rangle &= \langle d_{-n}^* d_{-n}v, v\rangle = \langle d_n d_{-n}v, v\rangle = \langle [d_n, d_{-n}]v, v\rangle \\&= 2n \langle d_0 v, v\rangle = 2n h.\) 

This implies in particular that \(h \ge 0\)  and that
\(h = 0\)  implies \(d_n v= 0\)  for each \(n \in {\mathbb {Z}}\) .
Now an easy direct calculation leads to
\( 0 \le \det \begin{pmatrix} \langle d_{-2n}v, d_{-2n}v \rangle &\langle d_{-n}^2v, d_{-2n}v \rangle \\\langle d_{-2n}v, d_{-n}^2v \rangle &\langle d_{-n}^2v, d_{-n}^2v \rangle  \end{pmatrix}= 4 n^3 h^2 (8h - 5n) \) 
([47]}; see also [48]}).
If \(h \ne 0\) , this expression is negative for sufficiently large \(n\) ,
so that we must have \(h = 0\) . This means that
\({\mathfrak {g}}.v = \lbrace 0\rbrace \) , and hence that \(v \in \mathcal {H}^G\)  is a fixed vector
(cf. [49]}).
The preceding argument implies that each semibounded
unitary representation \((\pi , \mathcal {H})\)  of \(G\)  on a non-zero
Hilbert space satisfies \(\mathcal {H}^G \ne \lbrace 0\rbrace \) .
Applying this to the representation on the invariant subspace
\((\mathcal {H}^G)^\bot \) , which is also semibounded, we find that this
space is trivial, and hence that \(\mathcal {H}= \mathcal {H}^G\) , i.e.,
the representation is trivial.
\(\Box \)
Remark 8.8  A smooth unitary representation
\((\pi , \mathcal {H})\)  of \(\mathop {{\rm Diff}}\nolimits ({\mathbb {S}}^1)_+\)  is said to be a positive
energy representation if the operator
\(-i{\tt d}\pi (\partial _\theta )\)  has non-negative spectrum.
This means that \(\partial _\theta \in I_\pi ^\star \) , so that
\(W_{\mathcal {V}({\mathbb {S}}^1)} \subseteq I_\pi ^\star \)  leads to \(W_{\mathcal {V}({\mathbb {S}}^1)} \subseteq C_\pi \) ,
and therefore \(\pi \)  is semibounded.
Hence the preceding theorem implies in particular that
all positive energy representations of \(\mathop {{\rm Diff}}\nolimits ({\mathbb {S}}^1)_+\)  are
trivial.

Problem 8.9  It would be nice to have an analog
of Theorem REF  for the universal covering
group \(\widetilde{G}\)  of \(G = \mathop {{\rm Diff}}\nolimits ({\mathbb {S}}^1)_+\) , which has the
fundamental group \(\pi _1(G) = {\mathbb {Z}}\) .
Then \(\partial _\theta \)  generates a subgroup
\(\widetilde{T} \subseteq \widetilde{G}\)  isomorphic to \({\mathbb {R}}\) . Since this group
is non-compact, we cannot expect it to have eigenvectors, so that
the argument in the proof of Theorem REF  does
not apply.
What we would need in this context is a suitable direct
integral decomposition with respect to the subgroup
\(Z := Z(\widetilde{G}) \cong {\mathbb {Z}}\) . If \((\pi , \mathcal {H})\)  is a semibounded
representation of \(\widetilde{G}\)  with \(\pi (Z) \subseteq \) , i.e.,
\(\pi \)  has a central character (which is a consequence of
Schur's Lemma if \(\pi \)  is irreducible), then
\({\rm Spec}({\tt d}\pi (d_0)) \subseteq \lambda + {\mathbb {Z}}\)  for some
\(\lambda \in {\mathbb {R}}\) , and the argument from above applies.
This proves that all irreducible semibounded representations
of \(\widetilde{G}\)  are trivial.
We expect that a general semibounded representation of \(\widetilde{G}\) 
has a direct integral decomposition
\(\mathcal {H}\cong \int ^\oplus _{\widehat{Z}} \mathcal {H}_\chi \, d\mu (\chi )\) 
with respect to some measure \(\mu \)  on \(\widehat{Z} \cong and that the semiboundedness of \)\( implies that all representations\)\( on the spaces \) H\( with central character\)\( are semibounded, hence trivial, and this would imply thatall semibounded representations of \) G\( are trivial.\)
Invariant cones in the Virasoro algebra
In the analytic context, the Virasoro algebra
is usually defined as the
central extension \(\mathop {{\mathfrak {vir}}}\nolimits = {\mathbb {R}}\oplus _{\omega _{GF}} \mathcal {V}({\mathbb {S}}^1)\) 
defined by the Gelfand–Fuchs cocycle
\(\omega _{GF}(f\partial _\theta , g\partial _\theta ):= \int _0^{2\pi } f^{\prime }g^{\prime \prime }\, d\theta = \frac{1}{2}\int _0^{2\pi } f^{\prime }g^{\prime \prime } - f^{\prime \prime }g^{\prime }\, d\theta = \int _0^{2\pi } f^{\prime \prime \prime }g\, d\theta .\) 
In many situations, the cohomologous cocycle
\(\omega (f\partial _\theta , g\partial _\theta )&:=& \int _0^{2\pi } (f^{\prime \prime \prime }+f^{\prime })g\, d\theta = \omega _{GF}(f\partial _\theta , g\partial _\theta )- {\textstyle {\frac{1}{2}}}\int _0^{2\pi } fg^{\prime } -f^{\prime }g\, d\theta \\&=& \omega _{GF}(f\partial _\theta , g\partial _\theta )- {\textstyle {\frac{1}{2}}}\lambda ([f\partial _\theta , g\partial _\theta ]),\) 
with \(\lambda (f\partial _\theta ) = \int _0^{2\pi }f\, d\theta \) ,
turns out to be more convenient.
Remark 8.10  
On the generators \(d_n = i e^{i n\theta } \partial _\theta \in \mathcal {V}({\mathbb {S}}^1)_ from (\ref {eq:wittgen}) we have\begin{equation}\omega (d_n, d_{-n}) = 2\pi i (n^3-n).\end{equation}With the central element \) c := (24 i,0) i virvir,
we thus obtain the relation
\( [d_n, d_m] = (n-m) d_{n+m} + \delta _{n,-m} \frac{n^3 -n}{12}\widehat{c} \) 
if we identify \(d_n\)  with the corresponding element
\((0,d_n) \in \mathop {{\mathfrak {vir}}}\nolimits _ (\cite [p.~9]{KR87}).\) Since we shall need them in the following, we record some related formulas.
First we observe that
\({\mathfrak {t}}:= {\mathbb {R}}c + {\mathbb {R}}d \quad \mbox{ with } \quad c := (1,0),d := (0,\partial _\theta )\) 
is a maximal abelian subalgebra of \(\mathop {{\mathfrak {vir}}}\nolimits \) .
The relation \([d_0, d_n] = - n d_n\)  implies that
\(d_n \in \mathop {{\mathfrak {vir}}}\nolimits _ is a root vectorfor the root \) n t*\( defined by\)\( \alpha _n(\widehat{c}) = 0 \quad \mbox{ and } \quad \alpha _n(d_0) = -n. \)\(In particular,\begin{equation}\alpha _n([d_n, d_n^*]) = \alpha _n([d_n, d_{-n}])= 2n \alpha _n(d_0) = - 2n^2 < 0 \quad \mbox{ for } \quad n \ne 0.\end{equation}We also observe that\begin{eqnarray} [d_n^*, d_n] = [d_{-n}, d_n]&= (\omega (d_{-n}, d_n), -2n d_0)= -i (2\pi (n^3 - n), 2n \partial _\theta )\\= - 2i n (\pi (n^2 - 1), \partial _\theta ).\end{eqnarray}\)
In view of the \(G\) -invariant pairing of the space \(\mathcal {F}_2\)  of 2-densities
with vector fields, the 1-cocycle \({\mathfrak {g}}\rightarrow {\mathfrak {g}}^{\prime }, x \mapsto i_x\omega \) 
corresponds to the 1-cocycle
\( {\mathfrak {g}}\rightarrow \mathcal {F}_2, \quad f \partial _\theta \mapsto (f^{\prime \prime \prime } + f^{\prime }) (d\theta )^2. \) 
To obtain a formula for the adjoint action of \(G = \mathop {{\rm Diff}}\nolimits ({\mathbb {S}}^1)_+^{\rm op}\) 
on \(\mathop {{\mathfrak {vir}}}\nolimits \) , we therefore need a group cocycle \(G \rightarrow \mathcal {F}_2\)  integrating
this 1-cocycle (Remark REF ).
Definition 8.11 
The Schwarzian derivative
\( S(\varphi ):= \frac{\varphi ^{\prime } \varphi ^{\prime \prime \prime } - \frac{3}{2} (\varphi ^{\prime \prime })^2}{(\varphi ^{\prime })^2}= \frac{\varphi ^{\prime \prime \prime }}{\varphi ^{\prime }} - \frac{3}{2} \Big (\frac{\varphi ^{\prime \prime }}{\varphi ^{\prime }}\Big )^2 \) 
assigns to \(\varphi \in G\)  a
\(2\pi \) -periodic smooth function. It satisfies the cocycle identity
\( S(\varphi \circ \psi ) = \big (S(\varphi ) \circ \psi \big )\cdot (\psi ^{\prime })^2 + S(\psi ) \) 
(cf. [42]}),
which means that it defines an \(\mathcal {F}_2\) -valued 1-cocycle
on \(G\) .

We easily derive that
\(T_{\mathop {{\rm id}}\nolimits }(S)(f) = f^{\prime \prime \prime },\)  and therefore the modified Schwarzian
derivative
\( \widetilde{S}(\varphi ) := S(\varphi ) + {\textstyle {\frac{1}{2}}}((\varphi ^{\prime })^2 - 1)= S(\varphi ) + \varphi .{\textstyle {\frac{1}{2}}}- {\textstyle {\frac{1}{2}}}\) 
is a cohomologous 1-cocycle with
\(T_{\mathop {{\rm id}}\nolimits }(\widetilde{S})(f) = f^{\prime \prime \prime } + f^{\prime }\)  (cf. [35]}). Therefore
Remark REF  implies that the coadjoint action
of \(G\)  on the smooth dual
\({\mathbb {R}}\times \mathcal {F}_2 \cong {\mathbb {R}}\times C^\infty ({\mathbb {S}}^1)\) 
of \(\mathop {{\mathfrak {vir}}}\nolimits \)  is given by
\(\mathop {{\rm Ad}}\nolimits ^*_{\varphi }(a, u)= \big (a, (u\circ \varphi ) (\varphi ^{\prime })^2 - a \widetilde{S}(\varphi )\big ),\) 
whereas, in view of (REF ), the adjoint action on \(\mathop {{\mathfrak {vir}}}\nolimits = {\mathbb {R}}\oplus _\omega \mathcal {V}({\mathbb {S}}^1) \cong {\mathbb {R}}\times C^\infty ({\mathbb {S}}^1)\) 
is given by
\( \mathop {{\rm Ad}}\nolimits _\varphi (z,f)= \Big (z - \int _0^{2\pi } f \widetilde{S}(\varphi ^{-1})\, d\theta , (f \circ \varphi )\cdot (\varphi ^{\prime })^{-1}\Big ). \) 
We are especially interested in the adjoint action
on the open convex cone
\( W_{\rm max} := \lbrace  (z,f) \in \mathop {{\mathfrak {vir}}}\nolimits \colon  f > 0\rbrace , \) 
which is the inverse image of the positive invariant cone \(W_{\mathcal {V}({\mathbb {S}}^1)}\subseteq \mathcal {V}({\mathbb {S}}^1)\) 
under the quotient map \(\mathop {{\mathfrak {vir}}}\nolimits \rightarrow \mathcal {V}({\mathbb {S}}^1), (z,f) \mapsto f\) .
From the corresponding results for \(W_{\mathcal {V}({\mathbb {S}}^1)}\) , we derive immediately
that each orbit in \(W_{\rm max}\)  intersects \({\mathfrak {t}}\) .
Proposition 8.12  
For each \((z,f) \in W_{\rm max}\) , the \(G\) -orbit of \((z,f)\) 
meets \({\mathfrak {t}}\)  in a unique element \((\beta (z,f),\alpha (z,f))\) , given by
\( \alpha (z,f) := \frac{1}{\chi (f)} \quad \mbox{ and } \quad \beta (z,f) := z - \int _0^{2\pi } \frac{(f^{\prime })^2}{2f}\, d\theta + {\textstyle {\frac{1}{2}}}\int _0^{2\pi } f\, d\theta - \frac{\pi }{\chi (f)}. \) 
For \(G = \mathop {{\rm Diff}}\nolimits ({\mathbb {S}}^1)_+\)  and \(T = \exp ({\mathbb {R}}\partial _\theta )\) , the
orbit map induces a diffeomorphism
\( \Gamma \colon  G/T \times (W_{\rm max} \cap {\mathfrak {t}}) \rightarrow W_{\rm max}, \quad (\varphi T,(\beta , \alpha )) \mapsto \mathop {{\rm Ad}}\nolimits _{\varphi }(\beta , \alpha ). \) 


Proof.   
Let \((z,f) \in W_{\rm max}\)  and assume that
\(\mathop {{\rm Ad}}\nolimits _{\varphi ^{-1}}(z,f) = (\beta , \alpha ) \in {\mathfrak {t}}\) , i.e.,
\( (\beta , \alpha )= \Big (z - \int _0^{2\pi } f\widetilde{S}(\varphi )\, d\theta , (f\cdot \varphi ^{\prime })\circ \varphi ^{-1}\Big ). \) 
Then \(\varphi ^{\prime } = \frac{\alpha }{f}\)  leads to \(\varphi ^{\prime \prime } = - \alpha \frac{f^{\prime }}{f^2}\) 
and
\( \varphi ^{\prime \prime \prime }= - \alpha \frac{f^{\prime \prime }}{f^2} + 2\alpha \frac{(f^{\prime })^2}{f^3}= \frac{\alpha }{f^3}(2 (f^{\prime })^2 - ff^{\prime \prime }). \) 
This leads to
\(S(\varphi ) = \frac{\varphi ^{\prime \prime \prime }}{\varphi ^{\prime }} - \frac{3}{2}\Big (\frac{\varphi ^{\prime \prime }}{\varphi ^{\prime }}\Big )^2&= \frac{1}{f^2}(2(f^{\prime })^2 - ff^{\prime \prime }) - \frac{3}{2}\frac{(f^{\prime })^2}{f^2}= \frac{1}{2f^2}((f^{\prime })^2 - 2ff^{\prime \prime })\\&= \frac{(f^{\prime })^2}{2f^2}- \frac{f^{\prime \prime }}{f},\) 

so that we obtain with \(\int _0^{2\pi } f^{\prime \prime }\, d\theta = f^{\prime }(2\pi ) - f^{\prime }(0) = 0\)  the relation
\( \int _0^{2\pi } f S(\varphi )\, d\theta =\int _0^{2\pi } \frac{(f^{\prime })^2}{2f}\, d\theta . \) 
Next we use \(\varphi ^{\prime } = \frac{\alpha }{f}\)  to obtain
\( {\textstyle {\frac{1}{2}}}\int _0^{2\pi } f ((\varphi ^{\prime })^2 -1)\, d\theta = {\textstyle {\frac{1}{2}}}\int _0^{2\pi } \alpha \varphi ^{\prime } - f \, d\theta = \pi \alpha - {\textstyle {\frac{1}{2}}}\int _0^{2\pi } f \, d\theta . \) 
Combining all that, we get
\(\beta &= z - \int _0^{2\pi } \frac{(f^{\prime })^2}{2f}\, d\theta + {\textstyle {\frac{1}{2}}}\int _0^{2\pi } f \, d\theta - \pi \alpha .\) 

We also obtain from \(\varphi ^{\prime } = \frac{\alpha }{f}\)  the relation
\(\chi (f)= \frac{1}{2\pi }\int _0^{2\pi } \frac{1}{f}\, d\theta = \frac{1}{\alpha },\) 
so that \(\alpha = \frac{1}{\chi (f)}\) .
This proves the first assertion.
Since \(T\)  fixes the subalgebra \({\mathfrak {t}}\)  pointwise,
\(\Gamma \)  is a well-defined smooth map. As a manifold, we may identify
\(G/T\)  with the set
\(\lbrace  \varphi ^{\prime } \colon  \varphi \in G \rbrace  = \Big \lbrace  h \in C^\infty ({\mathbb {S}}^1) \colon  h > 0,\int _0^{2\pi } h\, d\theta = 2 \pi \Big \rbrace . \) 
As we have seen above, the inverse of \(\Gamma \)  is given by
\( \Gamma ^{-1}(z,f) = \big (\varphi , (\beta (z,f), \alpha (z,f))\big ), \) 
where \(\varphi ^{\prime } = \frac{\alpha }{f} = \frac{1}{f \chi (f)}\) ,
and this map is also smooth.
Therefore \(\Gamma \)  is a diffeomorphism.
\(\Box \)
The function \(\beta \)  is rather complicated, but it can
be analyzed to some extent as follows. First we observe that
for the probability measure \(\mu = \frac{1}{2\pi } \, d\theta \) 
Jensen's inequality and the convexity of the function \(\frac{1}{x}\) 
on the positive half line imply that
\( \frac{1}{\int _0^{2\pi } f\, d\mu } \le \int _0^{2\pi } \frac{1}{f}\, d\mu = \chi (f) = \frac{1}{\alpha }, \) 
which implies that
\(\int _0^{2\pi } f\, d\theta \ge 2\pi \alpha .\) 
Therefore the sign of \(\beta (0,f)\)  is not clear at all,
but we shall see below that \(\beta (0,f) \le 0\) .
Lemma 8.13 

The function \(\beta \)  is concave.

Proof.   
To show that \(\beta \)  is concave, we have
to verify that \(\partial _h^2\beta \le 0\)  in each point of \(W_{\rm max}\) .
Since \(\beta \)  is \(\mathop {{\rm Ad}}\nolimits (G)\) -invariant and each orbit meets \({\mathfrak {t}}\) ,
it suffices to verify this in points \((z,f)\) , where
\(f\)  is constant, so that
\( (\partial _h^2 \beta )(z,f)= - f^{-1} \int _0^{2\pi }(h^{\prime })^2\, d\theta - \pi \partial _h^2(\chi ^{-1})(f). \) 
Further,
\( \partial _h(\chi ^{-1}) = - \chi ^{-2} \partial _h \chi \quad \mbox{ and } \quad \partial _h^2(\chi ^{-1}) = - \chi ^{-2} (\partial _h^2 \chi )+ 2 \chi ^{-3} (\partial _h \chi )^2, \) 
so that we obtain with \(\chi (f) = f^{-1}\)  (\(f\)  is constant) and
the formulas (REF ) in the
proof of Lemma REF  the relations
\( (\partial _h \chi )(f) = -\frac{1}{2\pi f^2}\int _0^{2\pi } h\, d\theta \quad \mbox{ and } \quad (\partial _h^2 \chi )(f) = \frac{1}{\pi f^3}\int _0^{2\pi } h^2\, d\theta . \) 
This leads further to
\(\partial _h^2(\chi ^{-1})(f)&= - \frac{f^2}{\pi f^3}\int _0^{2\pi } h^2\, d\theta + 2 f^3 \frac{1}{4\pi ^2f^4}\Big (\int _0^{2\pi } h\, d\theta \Big )^2 \\&= \frac{1}{\pi f}\Big (-\int _0^{2\pi } h^2\, d\theta + \frac{1}{2\pi }\Big (\int _0^{2\pi } h\, d\theta \Big )^2\Big ).\) 

Putting everything together, we arrive at
\( f \cdot (\partial _h^2 \beta )(z,f)= - \int _0^{2\pi } (h^{\prime })^2\, d\theta + \int _0^{2\pi } h^2\, d\theta - \frac{1}{2\pi }\Big (\int _0^{2\pi } h\, d\theta \Big )^2. \) 
We thus obtain a rotation invariant
quadratic form on \(C^\infty ({\mathbb {S}}^1)\) , so that it
is diagonal with respect to
Fourier expansion. Evaluating it in the basis functions
\(\cos (n\theta )\)  and \(\sin (n\theta )\)  immediately shows that it is
negative semidefinite.
\(\Box \)
Theorem 8.14  (Convexity Theorem for adjoint orbits of \(\mathop {{\mathfrak {vir}}}\nolimits \) )

For each \(x \in {\mathbb {R}}c + {\mathbb {R}}_+ \partial _\theta \subseteq {\mathfrak {t}},\)  we have
\( p_{\mathfrak {t}}(\mathcal {O}_x) \subseteq x + C_+ \quad \mbox{ for } \quad C_+ := {\mathbb {R}}_+ c + {\mathbb {R}}_+ \partial _\theta . \) 
If \(x\)  is not central, then we even have the equality
\( p_{\mathfrak {t}}(\overline{\mathop {{\rm conv}}\nolimits }(\mathcal {O}_x)) = x + C_+. \) 


Proof.   
By continuity of the projection \(p_{\mathfrak {t}}\) , it suffices
to assume that \(x = (\beta _0, \alpha _0)\)  with a constant \(\alpha _0 > 0\) , so that
\(x \in W_{\rm max} \cap {\mathfrak {t}}\) . Then \(\alpha (x) =\alpha _0\)  and
\(\beta (x) = \beta _0\) . Further,
\(p_{\mathfrak {t}}(\mathcal {O}_x) \subseteq \overline{\mathop {{\rm conv}}\nolimits }(\mathcal {O}_x)\) 
(Proposition REF ), so that the
convexity of the functions \(\chi = \alpha ^{-1}\)  and \(-\beta \) 
(Lemma REF ) implies that
for \((\widetilde{\beta }, \widetilde{\alpha }) \in p_{\mathfrak {t}}(\mathcal {O}_x)\) , we have
\( \widetilde{\alpha }\ge \alpha _0 \quad \mbox{ and } \quad \widetilde{\beta }\ge \beta _0. \) 
This means that \(p_{\mathfrak {t}}(\mathcal {O}_x) \subseteq x + C_+\) .
Now we assume that \(x = \beta c + \alpha \partial _\theta \)  with
\(\alpha > 0\) . In view of ()
in Remark REF ,
Proposition REF  implies that
\( p_{\mathfrak {t}}(\mathcal {O}_x)\supseteq x + {\mathbb {R}}^+ \alpha _n(x) [d_n^*, d_n]= x + {\mathbb {R}}^+ \alpha _n(-i x) i [d_n^*, d_n]. \) 
Further \(d_0 = i \partial _\theta \)  leads to
\(\alpha _n(-ix) > 0\) , so that
\( p_{\mathfrak {t}}(\mathcal {O}_x) \supseteq x + {\mathbb {R}}^+ i [d_n^*, d_n], \quad n \in {\mathbb {N}}. \) 
Next we recall from () that
\( i [d_n^*, d_n] = 2n (\pi (n^2 - 1), \partial _\theta )= 2n (\pi (n^2-1)c + d). \) 
For \(n = 1\)  we obtain \(2(0,\partial _\theta )\) , and for \(n \rightarrow \infty \) 
we have positive multiples of
\((1, \frac{1}{\pi (n^2-1)}\partial _\theta ) \rightarrow c\) ,
so that the closed convex cone generated by the elements
\(i[d_n^*, d_n]\)  is \(C_+ = {\mathbb {R}}_+ c + {\mathbb {R}}_+ d\) .
This proves that
\( p_{\mathfrak {t}}(\overline{\mathop {{\rm conv}}\nolimits }(\mathcal {O}_x))= \overline{\mathop {{\rm conv}}\nolimits }(\mathcal {O}_x) \cap {\mathfrak {t}}\supseteq x + C_+, \) 
and our proof is complete.
\(\Box \)
Note that the following theorem can not be derived from
the “general” Lemma REF  because
\({\mathfrak {t}}\)  is 2-dimensional.
Theorem 8.15  (Classification of open invariant cones in \(\mathop {{\mathfrak {vir}}}\nolimits \) )

The following statements classify the open invariant convex cones
in \(\mathop {{\mathfrak {vir}}}\nolimits \) :
(i)
Each proper open invariant convex cone in \(\mathop {{\mathfrak {vir}}}\nolimits \) 
is either contained in \(W_{\rm max}\)  or \(-W_{\rm max}\) .
(ii)
Each proper open invariant convex cone \(W \subseteq \mathop {{\mathfrak {vir}}}\nolimits \) 
is uniquely determined by
\(C := W \cap {\mathfrak {t}}\)  via \(W = \mathop {{\rm Ad}}\nolimits (G)C\) .
(iii)
Let \(C_{\rm max} := W_{\rm max} \cap {\mathfrak {t}}\) . Then an
open convex cone \(C \subseteq C_{\rm max}\) 
is the trace of an invariant open convex cone if and only if \(C_+^0 \subseteq C\) .
(iv)
If \(W_{\rm min}\)  is the open invariant cone corresponding
to \(C_{\rm min} := C_+^0\) , then each open invariant convex
cone \(W \subseteq W_{\rm max}\)  contains \(W_{\rm min}\) .
(v)
An open invariant convex cone \(W\)  is pointed if and only if \(C\)  is
pointed. In particular, \(W_{\rm min}\)  is pointed.


Proof.   
(i) Let \(W \subseteq \mathop {{\mathfrak {vir}}}\nolimits \)  be an open invariant convex cone.
If \({\mathfrak {z}}(\mathop {{\mathfrak {vir}}}\nolimits ) \cap W = \emptyset \) , then
\({\mathfrak {z}}(\mathop {{\mathfrak {vir}}}\nolimits ) + W\)  is a proper cone,
and therefore its image in \(\mathcal {V}({\mathbb {S}}^1)\)  is contained in \(W_{\mathcal {V}({\mathbb {S}}^1)}\)  or
\(-W_{\mathcal {V}({\mathbb {S}}^1)}\) ,
so that \(W \subseteq W_{\rm max}\)  or \(W \subseteq - W_{\rm max}\) 
(Theorem REF ).
To verify (i), we therefore have to show that
\(c \notin C := W \cap {\mathfrak {t}}\) . Suppose the converse.
Then there exists an \(\varepsilon > 0\)  with
\(x_\pm := c \pm \varepsilon \partial _\theta \in C\) .
From Theorem REF 
we derive that
\( p_{\mathfrak {t}}(\overline{\mathop {{\rm conv}}\nolimits }(\mathcal {O}_{x_+})) = x_+ + C_+, \) 
and, applying it also to \(-x_-\) , we find that
\( p_{\mathfrak {t}}(\overline{\mathop {{\rm conv}}\nolimits }(\mathcal {O}_{x_-})) = x_- - C_+. \) 
Since both sets are contained in \(\overline{C}\) , we see that
\(\pm C_+ \subseteq \lim (\overline{C}) = \lim (C)\) 
(Lemma REF ), so that \(\lim (C) = {\mathfrak {t}}\) , and thus
\(0 \in C = {\mathfrak {t}}\) . As \(W\)  is open, \(0 \in W\)  leads to
\(W = \mathop {{\mathfrak {vir}}}\nolimits \) .
(ii) follows from (i) and Proposition REF .
(iii) If \(C = W \cap {\mathfrak {t}}\)  for an invariant open convex cone \(W\) ,
then \(C \subseteq C_{\rm max}\)  implies that \(C \cap {\mathfrak {z}}(\mathop {{\mathfrak {vir}}}\nolimits ) = \lbrace 0\rbrace \) .
Therefore we have for \(x \in C\)  the relation
\( p_{\mathfrak {t}}(\overline{\mathop {{\rm conv}}\nolimits }(\mathcal {O}_x)) = x + C_+ \subseteq \overline{C}, \) 
and thus \(C_+ \subseteq \lim (\overline{C}) = \overline{C}\) ,
which in turn yields \(C_+^0 \subseteq C\) .
If, conversely, \(C_+^0 \subseteq C\)  holds for an open convex cone
\(C \subseteq C_{\rm max}\) ,
then \(C_+ \subseteq \lim (C) = \overline{C}\)  leads for each \(x \in C\)  to
\( p_{\mathfrak {t}}(\mathcal {O}_x) \subseteq x + C_+ \subseteq C. \) 
Therefore
\( W_C:= \lbrace  x \in \mathop {{\mathfrak {vir}}}\nolimits \colon  p_{\mathfrak {t}}(\mathcal {O}_x) \subseteq C \rbrace = \bigcap _{\varphi \in G} \varphi .p_{\mathfrak {t}}^{-1}(C) \) 
is a convex invariant cone containing the subset
\(\mathop {{\rm Ad}}\nolimits (G)C\)  which is open by Proposition REF .
Hence \(W_C^0\)  is an open invariant convex cone satisfying \(W_C^0 \cap {\mathfrak {t}}=~C\) .
(iv) follows immediately from (iii).
(v) If \(W\)  is pointed, i.e., it contains no affine lines,
then the same holds for \(C := W \cap {\mathfrak {t}}\) . If, conversely,
\(C\)  contains no affine lines, then
\(H(W)\)  is a closed ideal of \(\mathop {{\mathfrak {vir}}}\nolimits \)  intersecting
\({\mathfrak {t}}\)  trivially. Hence it is contained in
\([{\mathfrak {t}}, \mathop {{\mathfrak {vir}}}\nolimits ]\)  and \(H(W)_ is adapted to the rootdecomposition with respect to \) t. If it contains
\(d_n\) , then its \(*\) -invariance implies that it also contains \(d_n^* = d_{-n}\) ,
which leads to the contradiction \([d_n, d_{-n}] \in H(W)_.This implies that \) H(W) = {0}\(, so that \) W\( is pointed.\hfill \Box \)
As a consequence of the preceding theorem, the cones
\(W_{\rm min}\) , resp., \(W_{\rm max}\)  play the role of a minimal, resp.,
maximal open invariant cone in \(\mathop {{\mathfrak {vir}}}\nolimits \) . The existence of minimal
and maximal invariant cones is a well known phenomenon for
finite dimensional hermitian Lie algebras (cf. [17]} and
Proposition REF ).
Corollary 8.16  
The smallest closed convex invariant cone in \(\mathop {{\mathfrak {vir}}}\nolimits \)  containing
\(\partial _\theta \)  is the closure of \(W_{\rm min}\) .

Proof.   
If \(D \subseteq \mathop {{\mathfrak {vir}}}\nolimits \)  is a closed convex invariant cone,
then \(\partial _\theta \in D\)  implies that
\(\partial _\theta + C_+ \subseteq D\) 
(Theorem REF ), so that
\(C_+ \subseteq D\)  (Lemma REF ),
and thus \(W_{\rm min} \subseteq D\)  by invariance.
\(\Box \)
Remark 8.17  In view of the preceding theorem, the open invariant convex
cones in \(\mathop {{\mathfrak {vir}}}\nolimits \)  can be classified as follows. Since the closure of the
cone \(C\)  contains
\(C_+ = {\mathbb {R}}_+ c + {\mathbb {R}}_+ \partial _\theta \)  and is contained in
\(\overline{C_{\rm max}} = {\mathbb {R}}c + {\mathbb {R}}_+ \partial _\theta \) ,
we have \(\overline{C} = {\mathbb {R}}_+ c + {\mathbb {R}}_+ (\partial _\theta - \alpha c)\) 
for some \(\alpha > 0\)  whenever \(C \ne  C_{\rm min}, C_{\rm max}\) .

Semi-equicontinuity of coadjoint orbits of \(\mathop {{\mathfrak {vir}}}\nolimits ^{\prime }\)
In this final section on the Virasoro algebra we apply the
detailed results on invariant cones to semibounded representations
and semi-equicontinuous coadjoint orbits.
In particular, we show that the set \({\mathfrak {g}}^{\prime }_{\rm seq}\)  of semi-equicontinuous
coadjoint orbits coincides with
the double cone \(W_{\rm min}^\star \cup -W_{\rm min}^\star \) .
This in turn is used to show that the unitary highest weight
representations of the Virasoro group are precisely the
irreducible semibounded representations
and to determine their momentum sets.
Proposition 8.18  For \(\lambda \in \mathop {{\mathfrak {vir}}}\nolimits ^{\prime }\)  and \(d = (0,\partial _\theta )\) ,
the following
are equivalent:
(i)
\(\mathcal {O}_\lambda \)  is semi-equicontinuous.
(ii)
The convex cone \(B(\mathcal {O}_\lambda )\)  contains
\(W_{\rm max}\)  or \(-W_{\rm max}\) .
(iii)
\(\mathcal {O}_\lambda (d) = \lambda (\mathcal {O}_d)\)  is bounded from
below or above.
(iv)
\(\lambda \in W_{\rm min}^\star \cup -W_{\rm min}^\star \) .
(v)
\(\mathcal {O}_\lambda (d) \ge 0\)  or \(\lambda (\mathcal {O}_d) \le 0\) .


Proof.   
(i) \(\Rightarrow \)  (ii): If \(\mathcal {O}_\lambda \)  is semi-equicontinuous,
then the invariant convex cone \(B(\mathcal {O}_\lambda )\)  has interior points
and contains \({\mathfrak {z}}(\mathop {{\mathfrak {vir}}}\nolimits )\) . Therefore
\(B(\mathcal {O}_\lambda )^0/{\mathfrak {z}}(\mathop {{\mathfrak {vir}}}\nolimits )\)  is an open invariant convex cone
in \(\mathcal {V}({\mathbb {S}}^1)\) , hence contains \(W_{\mathcal {V}({\mathbb {S}}^1)}\)  or \(-W_{\mathcal {V}({\mathbb {S}}^1)}\) 
(Theorem REF ).
This in turn implies that \(B(\mathcal {O}_\lambda )\)  contains either
\(W_{\rm max}\)  or \(-W_{\rm max}\) .
(ii) \(\Rightarrow \)  (i) follows from Proposition REF 
because \(\mathop {{\mathfrak {vir}}}\nolimits \)  is a Fréchet space.
(ii) \(\Rightarrow \)  (iii) follows from \(d \in W_{\rm max}\) .
(iii) \(\Rightarrow \)  (ii) follows from \({\mathbb {R}}c \subseteq B(\mathcal {O}_\lambda )\) 
and \(W_{\rm max} = \mathop {{\rm Ad}}\nolimits (G)({\mathbb {R}}c + {\mathbb {R}}_+^\times d)\) 
(Proposition REF ).
(i) \(\Rightarrow \)  (iv): If \(\lambda (c) = 0\) , then \(\mathcal {O}_\lambda \)  can be
identified with a semi-equicontinuous coadjoint orbit of \(\mathcal {V}({\mathbb {S}}^1)\) ,
so that Proposition REF  implies
that \(\lambda \in W_{\mathcal {V}({\mathbb {S}}^1)}^\star \cup - W_{\mathcal {V}({\mathbb {S}}^1)}^\star \) 
in \(\mathcal {V}({\mathbb {S}}^1)^{\prime }\) ,
which means that \(\lambda \in W_{\rm max}^\star \cup - W_{\rm max}^\star \)  in
\(\mathop {{\mathfrak {vir}}}\nolimits ^{\prime }\) .
If \(\lambda (c) \ne 0\) , then \(\mathcal {O}_\lambda \)  is contained
in the closed invariant hyperplane \(\lambda + c^\bot \) , so that
the construction in Remark REF  implies that the cone
\(\mathcal {O}_\lambda ^\star \)  has interior points.
Clearly, this cone is proper, so that Theorem REF 
implies that it either contains \(W_{\rm min}\)  or \(-W_{\rm min}\) ,
which in turn leads to \(\lambda \in W_{\rm min}^\star \cup - W_{\rm min}^\star \) .
(iv) \(\Rightarrow \)  (i): Since \(\pm W_{\rm min}\)  are open invariant cones,
their duals are semi-equicontinuous sets (Example REF (b)).
(iv) \(\Leftrightarrow \)  (v): Since the closed convex invariant cone generated
by \(d\)  is \(W_{\rm min}\)  (Corollary REF ),
\(\mathcal {O}_\lambda (d) \ge 0\)  is equivalent to
\(\lambda \in W_{\rm min}^\star .\) 
\(\Box \)
For any \(\lambda \in {\mathfrak {t}}^* \cong [{\mathfrak {t}},{\mathfrak {g}}]^\bot \subseteq {\mathfrak {g}}^{\prime }\) , the fact that
\(\mathcal {O}_\lambda \)  is constant on the central element
\(c= (1,0)\)  implies that
\(p_{{\mathfrak {t}}^*}(\mathcal {O}_\lambda ) \subseteq {\mathfrak {t}}^*\)  is a connected
subset of the affine line
\(\lbrace  \mu \in {\mathfrak {t}}^* \colon  \mu (c) = \lambda (c)\rbrace = \lambda + ({\mathfrak {t}}^* \cap c^\bot ). \) 
In particular, this set is convex.
Proposition 8.19  
If \(\lambda \in {\mathfrak {t}}^*\) , then
(a)
\(\mathcal {O}_\lambda (d)\)  is bounded from below
if and only if \(\lambda (d)\ge 0\)  and \(\lambda (c) \ge 0\) .
If this is the case and \(\lambda \ne 0\) , then
\( B(\mathcal {O}_\lambda )^0 = W_{\rm max} \quad \mbox{ and } \quad W_{\rm min} \subseteq \mathcal {O}_\lambda ^\star . \) 

(b)
\(p_{{\mathfrak {t}}^*}(\mathcal {O}_\lambda )\)  is contained in an affine
half-line if and only if \(\mathcal {O}_\lambda \)  is semi-equicontinuous
if and only
if \(\lambda (c)\lambda (d) \ge 0\) .


Proof.   
(a) We recall from Theorem REF  that
\(p_{{\mathfrak {t}}}(\overline{\mathop {{\rm conv}}\nolimits }(\mathcal {O}_d)) = d + C_+.\) 
This implies that
\(\lambda (\mathcal {O}_d)\)  is bounded from below if and only if
\(\lambda \in (C_+)^\star \) , i.e.,
\(\lambda (c), \lambda (d) \ge 0\) .
Suppose that these conditions are satisfied and that \(\lambda \ne 0\) .
Then \(B(\mathcal {O}_\lambda )^0\)  is a proper open invariant cone, hence
determined by its intersection with \({\mathfrak {t}}\)  (Theorem REF ).
As this intersection
contains \(d\)  and is invariant under translation with \({\mathbb {R}}c\) ,
Theorem REF  implies that it coincides with
\(C_{\rm max}\) . This proves that \(W_{\rm max} = B(\mathcal {O}_\lambda )^0\) .
We have already seen above that \(\lambda \in C_+^\star \)  and
since \(p_{{\mathfrak {t}}^*}(\mathcal {O}_\lambda )\)  is a half-line constant
on \(c\)  and bounded below on \(d\) ,
it follows that \(C_+ \subseteq \mathcal {O}_\lambda ^\star \) , which
leads to \(W_{\rm min} \subseteq \mathcal {O}_\lambda ^\star \) .
(b) We use Proposition REF  to obtain with the notation
of Remark  REF
\(p_{{\mathfrak {t}}^*}(\mathcal {O}_\lambda ) \supseteq \lambda + {\mathbb {R}}^+ \lambda ([d_n^*, d_n]) \alpha _n. \) 
We also know that
\(i [d_n^*, d_n] = 2n (\pi (n^2 - 1), \partial _\theta ),\) 
so that, for each \(n \in {\mathbb {N}}\) ,
\(p_{{\mathfrak {t}}^*}(\mathcal {O}_\lambda ) \supseteq \lambda - {\mathbb {R}}^+ \lambda (\pi (n^2-1), \partial _\theta ) i \alpha _n= \lambda - {\mathbb {R}}^+ \lambda (\pi (n^2-1), \partial _\theta ) i \alpha _1. \) 
If \(p_{{\mathfrak {t}}^*}(\mathcal {O}_\lambda )\)  is contained in a half-line, the signs
of the numbers
\( \lambda (\pi (n^2-1), \partial _\theta ), \quad n \in {\mathbb {N}}, \) 
have to coincide, which is equivalent to
\(\lambda (d) \lambda (c) \ge 0.\) 
If, conversely, this condition is satisfied,
then (a) implies that \(\mathcal {O}_\lambda (d)\)  is semibounded,
so that \(p_{{\mathfrak {t}}^*}(\mathcal {O}_\lambda )\)  is contained
in an affine half-line.
\(\Box \)
Definition 8.20  We write \(\mathop {{\rm Vir}}\nolimits \)  for the (up to isomorphism unique)
simply connected Lie group with Lie algeba \(\mathop {{\mathfrak {vir}}}\nolimits \) .

A unitary representation \((\pi , \mathcal {H})\)  of \(\mathop {{\rm Vir}}\nolimits \) 
is called a highest weight representation
if there exists a smooth cyclic vector \(0 \ne v \in \mathcal {H}^\infty \) 
which is a \({\mathfrak {t}}\) -eigenvector annihilated by each \(d_n\) , \(n > 0\) .
Then the corresponding eigenfunctional \(\lambda \in i{\mathfrak {t}}^*\)  is called
the highest weight and \(v\)  a highest weight vector.

Theorem 8.21  
If \((\pi _\lambda , \mathcal {H}_\lambda )\)  is a unitary
highest weight representation of the simply connected
Lie group \(\mathop {{\rm Vir}}\nolimits \)  with Lie algebra \(\mathop {{\mathfrak {vir}}}\nolimits \)  of highest weight
\(\lambda \in i {\mathfrak {t}}^*\) , then
\(\lambda (d_0) \ge 0 \quad \mbox{ and } \quad \lambda (\widehat{c}) \ge 0,\) 
i.e., \(i \lambda (\partial _\theta ) \ge 0\) 
and \(i\lambda (c) \ge 0.\) 
The representation \((\pi _\lambda , \mathcal {H}_\lambda )\) 
is semibounded and its momentum set is given by
\( I_{\pi _\lambda } = \overline{\mathop {{\rm conv}}\nolimits }(\mathcal {O}_{-i\lambda }). \) 
For \(\lambda \ne 0\)  we have
\( -W_{\rm max} = W_{\pi _\lambda } \quad \mbox{ and } \quad -W_{\rm min} \subseteq C_{\pi _\lambda }. \) 


Proof.   
First we use [47]} to see that the
unitarity of the irreducible highest weight module \(L(\lambda )\)  of \(\mathop {{\mathfrak {vir}}}\nolimits _with highest weight\) it*\( implies (\ref {eq:hiweisign}).Actually, this follows from the simple observation thatif \) v\( is a highest weight vector of unit length, then{\begin{@align*}{1}{-1}0 &\le \langle d_{-n}v_\lambda , d_{-n}v_\lambda \rangle = \langle d_{-n}^*d_{-n}v_\lambda , v_\lambda \rangle = \langle [d_{-n}^*, d_{-n}]v_\lambda , v_\lambda \rangle \\&= \lambda ([d_n, d_{-n}])= \lambda \Big (2n d_0 + \frac{n^3 - n}{12}\widehat{c}\Big ) \ge 0\end{@align*}}holds for each \) n N\(.\) The existence of a corresponding continuous unitary highest weight
representation \((\pi _\lambda , \mathcal {H}_\lambda )\)  of the simply connected
Lie group \(\mathop {{\rm Vir}}\nolimits \)  with Lie algebra \(\mathop {{\mathfrak {vir}}}\nolimits \)  has been shown by Goodman
and Wallach [54]}. A more general method of integration
which applies in particular to highest weight modules of \(\mathop {{\rm Vir}}\nolimits \) 
has been developed by Toledano Laredo ([55]}).
It is based on techniques related to regular Lie groups,
and [55]} implies in particular that the
highest weight vector \(v_\lambda \)  is smooth.
This vector is an eigenvector for the closed subalgebra
of \(\mathop {{\mathfrak {vir}}}\nolimits _ generated by \) t and the
\(d_n\) , \(n > 0\) .
Next we recall from Theorem REF  the complex manifold
\( M = \mathop {{\rm Diff}}\nolimits ({\mathbb {S}}^1)_+/\exp ({\mathbb {R}}\partial _\theta )\cong \mathop {{\rm Vir}}\nolimits /T, \) 
where \(T := \exp {\mathfrak {t}}\subseteq \mathop {{\rm Vir}}\nolimits \)  is the subgroup corresponding to
\({\mathfrak {t}}\) . Since the tangent space in the base point
\(x_0 = \mathbf {1} T\)  can be identified with
\(\mathop {{\mathfrak {vir}}}\nolimits _{\mathfrak {p}}\) , \({\mathfrak {p}}:= \overline{{\mathfrak {t}}_ \sum _{n < 0} d̏_n}\) ,
Theorem REF  implies that the
map
\(\eta \colon  M \rightarrow {\mathbb {P}}(\mathcal {H}_\lambda ^{\prime }),\quad gT \mapsto [\pi ^*_\lambda (g)\alpha _{v_\lambda }] \) 
is holomorphic. As \(v_\lambda \)  is cyclic, we thus obtain a
realization of the unitary representation \((\pi _\lambda , \mathcal {H}_\lambda )\) 
in the space of holomorphic sections of the holomorphic line bundle
\(\eta ^*{\mathbb {L}}_{\mathcal {H}_\lambda ^{\prime }}\)  over
\(M\)  (cf. Theorem REF )An infinitesimal version of this construction
can already be found in [57]} and [58]} contains
various formal aspects of the realization of the highest weight
representations in spaces of holomorphic functions, resp.,
sections of holomorphic line bundles on \(M\) .
From the highest weight structure of \(\mathcal {H}_\lambda \)  it follows that the
set of \({\mathfrak {t}}_-weights on \) H\( is given by\) - N0 1\(, so that \) i d()
= d(d0)\( is bounded from below. Therefore thecone \) Cmax\( from the Classification Theorem~\ref {thm:classinvcon-vir}satisfies \) - Cmax W\(,which immediately leads to \) - Wmax W\(.Since \) W\(,this cone is proper, and the Classification Theorem thus impliesthe equality \) W = -Wmax\(.With this information we now apply Theorem~\ref {thm:2.7}to determine the precise momentum set.\) From Theorem REF  it now follows that,
for each \(x \in W_{\rm max} = - W_{\pi _\lambda } = W_{\pi _\lambda ^*}\) ,
the smooth action of the corresponding one-parameter group
on \(M\)  extends to a holomorphic action of \(+\) .
Therefore Theorem REF (c) implies that
\(I_{\pi _\lambda }\)  is the closed convex hull of the coadjoint
orbit \(\Phi _{\pi _\lambda }(G[v_\lambda ])\) . In view of
\(\Phi _{\pi _\lambda }([v_\lambda ]) = -i\lambda \in {\mathfrak {t}}^* \subseteq \mathop {{\mathfrak {vir}}}\nolimits ^{\prime }\) ,
this proves that \(I_{\pi _\lambda } = \overline{\mathop {{\rm conv}}\nolimits }(\mathcal {O}_{-i\lambda })\) .
Now
\(C_{\pi _\lambda } = (I_{\pi _\lambda }^\star )^0 =(\mathcal {O}_{-i\lambda }^\star )^0 \supseteq -W_{\rm min} \) 
follows directly from Proposition REF ,
and since \(W_{\rm min}\)  has interior points,
\((\pi _\lambda , \mathcal {H}_\lambda )\)  is semibounded.
\(\Box \)
For more details on the classification of unitary highest
weight modules of \(\mathop {{\mathfrak {vir}}}\nolimits \) , we refer to [47]}.
We conclude this section with the following converse of
Theorem REF :
Theorem 8.22  
Every irreducible semibounded representation
\((\pi , \mathcal {H})\)  of \(\mathop {{\rm Vir}}\nolimits \)  is either a highest weight representation
or the dual of a highest weight representation.

Proof.   
We assume that the representation
\(\pi \)  is non-trivial, so that \(I_\pi \ne  \lbrace 0\rbrace \) .
Let \(p_{{\mathfrak {t}}^*} \colon  \mathop {{\mathfrak {vir}}}\nolimits ^{\prime } \rightarrow {\mathfrak {t}}^*\)  be the restriction map.
If \(p_{{\mathfrak {t}}^*}(I_\pi )= \lbrace 0\rbrace \) , then
\({\mathfrak {t}}\subseteq I_\pi ^\bot = \ker {\tt d}\pi \)  and thus
\(\mathop {{\mathfrak {vir}}}\nolimits = {\mathfrak {t}}+ [{\mathfrak {t}},\mathop {{\mathfrak {vir}}}\nolimits ] \subseteq \ker {\tt d}\pi \) , contradicting the
non-triviality of the representation.
Next we use Proposition REF 
to conclude that \(\lbrace 0\rbrace  \ne  p_{{\mathfrak {t}}^*}(I_\pi )\subseteq I_\pi \) .
Then each non-zero \(\lambda \in {\mathfrak {t}}^* \cap I_\pi \)  has a
semi-equicontinuous orbit, so that
\(\mathcal {O}_\lambda (d)\)  is bounded from below or above, and
in this case \(B(\mathcal {O}_\lambda )^0 = W_{\rm max}\)  or \(-W_{\rm max}\) 
(Proposition REF ). This implies in particular
that \(W_\pi \ne \mathop {{\mathfrak {vir}}}\nolimits \) , i.e., \(\pi \)  is not bounded.
As the open invariant cone \(W_\pi \)  is proper,
Theorem REF (i)
implies that \(W_\pi \)  is either
contained in \(W_{\rm max}\)  or \(-W_{\rm max}\) . We assume the latter
and claim that \(\pi \)  is a highest weight representation.
In the other case,
\(W_{\pi ^*} = - W_\pi \subseteq - W_{\rm max}\) , so that \(\pi \)  is the
dual of a highest weight representation.
First we note that \(W_\pi \subseteq - W_{\rm max}\)  implies
\(-W_{\rm min} \subseteq W_\pi \) .
As \(c \in H(W_\pi )\)  follows from \({\tt d}\pi (c) \in i {\mathbb {R}}\mathbf {1}\)  (Schur's Lemma),
we obtain the relation \({\mathbb {R}}c - W_{\min } \subseteq W_\pi \) , and therefore
\(W_{\rm max} = W_{\rm min} + {\mathbb {R}}c\)  leads to
\(W_\pi = - W_{\rm max}\ni - d\) .
Hence the spectrum of \(i{\tt d}\pi (d)= {\tt d}\pi (d_0)\)  is bounded from below.
In view of \(\exp (2\pi d) \in Z(\mathop {{\rm Vir}}\nolimits )\) ,
Proposition REF  implies the
existence of a smooth unit vector \(v \in \mathcal {H}^\infty \)  which is an
eigenvector for the minimal eigenvalue \(h\)  of \(i{\tt d}\pi (d)\) .
Then \({\mathfrak {t}}= {\mathbb {R}}c + {\mathbb {R}}d\) , \(v\)  is a \({\mathfrak {t}}\) -eigenvector and
\([d_0, d_n] = - n d_n\)  implies
that \(d_n.v = 0\)  for each \(n > 0\) .
Therefore \((\pi , \mathcal {H})\)  is a highest weight representation.
\(\Box \)
Symplectic group and metaplectic representation
In this section we study the metaplectic representation
\((\pi _s, S(\mathcal {H}))\)  of the
central extension \(\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\)  of \(\mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})\) 
on the symmetric Fock space
\(S(\mathcal {H})\) . This representation arises from self-intertwining
operators of the Fock representation of the Heisenberg group
\(\mathop {{\rm Heis}}\nolimits (\mathcal {H})\) . We show that it is semibounded and
determine the corresponding cone \(W_{\pi _s}\) .
For the larger central extension
\(\widehat{\mathop {\rm HSp{}}\nolimits }_{\rm res}(\mathcal {H})\)  of \(\mathcal {H}\rtimes \mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})\) 
the representation on \(S(\mathcal {H})\)  is irreducible
and semibounded and we show that its momentum set
is the weak-\(*\) -closed convex hull of a single coadjoint orbit.
The metaplectic representation
On the dense subspace \(S(\mathcal {H})_0 = \sum _{n = 0}^\infty S^n(\mathcal {H})\)  of the
symmetric Fock space \(S(\mathcal {H})\)  (cf. Appendix )
we have for each \(f \in \mathcal {H}\)  the creation operator
\( a^*(f)(f_1 \vee \cdots \vee f_n) := f \vee f_1 \vee \cdots \vee f_n. \) 
This operator has an adjoint \(a(f)\)  on \(S(\mathcal {H})_0\) , given by
\( a(f)\Omega = 0, \quad a(f)(f_1 \vee \cdots \vee f_n)= \sum _{j = 0}^n \langle f_j, f \rangle f_1 \vee \cdots \vee \widehat{f_j} \vee \cdots \vee f_n, \) 
where \(\widehat{f_j}\)  means omitting the factor \(f_j\) . Note that
\(a(f)\)  defines a derivation on the algebra \(S(\mathcal {H})_0\) .
One easily verifies
that these operators satisfy the
canonical commutation relations (CCR):
\([a(f), a(g)] = 0, \quad [a(f), a^*(g)] = \langle g, f\rangle \mathbf {1}.\) 
For each \(f \in \mathcal {H}\) , the operator \(a(f) + a^*(f)\)  on \(S(\mathcal {H})_0\) 
is essentially self-adjoint ([37]}), so that
\( W(f) := e^{\frac{i}{\sqrt{2}}\overline{a(f) + a^*(f)}} \in \mathop {\rm U{}}\nolimits (\mathcal {H}) \) 
is a unitary operator. These operators satisfy the Weyl
relations
\( W(f) W(f^{\prime }) = W(f + f^{\prime }) e^{\frac{i}{2}\mathop {{\rm Im}}\nolimits \langle f, f^{\prime }\rangle },\quad f,f^{\prime } \in \mathcal {H}. \) 
For the Heisenberg group
\( \mathop {{\rm Heis}}\nolimits (\mathcal {H}) := {\mathbb {R}}\times \mathcal {H}, \) 
with the multiplication
\( (t,v) (t^{\prime },v^{\prime }) := (t + t^{\prime } + {\textstyle {\frac{1}{2}}}\omega (v,v^{\prime }), v + v^{\prime }), \quad \omega (v,v^{\prime }) := \mathop {{\rm Im}}\nolimits \langle v,v^{\prime }\rangle \) 
we thus obtain by \(W(t, f) := e^{it} W(f)\) 
a unitary representation on \(S(\mathcal {H})\) , called the Fock representation.
It is a continuous irreducible representation
([37]}). That it actually is smooth
follows from the smoothness of
\(\langle W(t,f)\Omega , \Omega \rangle = e^{it - \frac{1}{4}\Vert f\Vert ^2}\) 
and Theorem REF .
Definition 9.1  The symplectic group \(\mathop {{\rm Sp}}\nolimits (\mathcal {H})\)  acts via
\(g.(t,v) := (t,gv)\)  by automorphism on the Heisenberg group,
and since the unitary representation
\((W,S(\mathcal {H}))\)  is irreducible, there exists for each
\(g \in \mathop {{\rm Sp}}\nolimits (\mathcal {H})\) , up to multiplication with \(, at most one unitaryoperator \) s(g) U(S(H))\( with\begin{equation}\pi _s(g) W(t,f) \pi _s(g)^* = W(t,gf) \quad \mbox{ for } \quad t\in {\mathbb {R}}, f \in \mathcal {H}.\end{equation}According to \cite {Sh62}, such an operator existsif and only if \) g Spres(H)\( (cf.\ Example~\ref {ex:liegrp}(d)),which immediatelyleads to a projective unitary representation\)\( \overline{\pi }_s \colon  \mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H}) \rightarrow \mathop {\rm PU{}}\nolimits (S(\mathcal {H})), \)\(determined by (\ref {eq:autrel}) for any lift\) s(g)U(S(H))\( of \)s(g)\(.Writing \)u\( for the image of \) u U(H)\( in the projectiveunitary group \) PU(H)\(, the corresponding pull-back\)\( \widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H}) := \overline{\pi }_s^*\mathop {\rm U{}}\nolimits (S(\mathcal {H}))= \lbrace  (g,u) \in \mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H}) \times \mathop {\rm U{}}\nolimits (S(\mathcal {H})) \colon \overline{u} = \overline{\pi }_s(g)\rbrace \)\(is called the {\it metaplectic group}.\glossary {name={\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})},description={metaplectic group}}\glossary {name={(\pi _s, S(\mathcal {H}))},description={metaplectic representation of metapl. group}}It is a centralextension of \) Spres(H)\( by \) . We shall
see below that this group is a Lie group and that
its canonical representation \(\pi _s(g,u) = u\) 
on \(S(\mathcal {H})\) , the metaplectic
representation, is smooth and semibounded. Our strategy
is to use Theorem REF ,
which requires a suitable lift of \(\overline{\pi }_s\) .

Remark 9.2  
From the relation \({\tt d}W(f) = \frac{i}{\sqrt{2}}(a(f) + a^*(f))\) ,
we recover the antilinear and linear part of \({\tt d}W\)  by
\( a(f) = \frac{1}{i\sqrt{2}} ({\tt d}W(f) + i {\tt d}W(If)), \quad a^*(f) = \frac{1}{i\sqrt{2}}({\tt d}W(f) - i {\tt d}W(If)), \) 
so that by ()
\( \pi _s(g) a(f) \pi _s(g)^{-1}= \frac{1}{i \sqrt{2}}\big ({\tt d}W(gf) + i {\tt d}W(gIf)\big )= a(g_1 f) + a^*(g_2 f) =: a_g(f), \) 
where \(g = g_1 + g_2\)  is the decomposition into linear an antilinear
part.

Theorem 9.3 
The topological group \(\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\)  is a Lie group and
the metaplectic representation is smooth.
A Lie algebra cocycle \(\eta \)  defining \(\widehat{\mathfrak {sp}}_{\rm res}(\mathcal {H})\) 
as an extension of \(\mathfrak {sp}_{\rm res}(\mathcal {H})\)  by \({\mathbb {R}}\)  is given by
\( \eta (x,y) = \frac{1}{2i} \mathop {{\rm tr}}\nolimits ([x_2, y_2]), \) 
where \(x_2\)  denotes the antilinear component of \(x\) .

Proof.   
Step 1: We consider the unbounded operators \(a(f)\)  and \(a^*(f)\)  on
\(S(\mathcal {H})_0 \subseteq S(\mathcal {H})\)  and observe that \(\)  is the common kernel
of the annihilation operators \(a(f)\) . For \(g\in \mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})\) ,
we observe that \(\pi _s(g)\Omega \)  lies in the common kernel of the
operators \(a_g(f) = \pi _s(g) a(f) \pi _s(g)^{-1}\) 
(cf. Remark REF ).
If \(g_1\)  is invertible, which is the case in some open
\(\mathbf {1}\) -neighborhood in \(\mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})\)  (actually on the whole group),
we consider the antilinear operator \(T(g) := g_2 g_1^{-1}\) 
for which \(a_g(g_1^{-1} f) = a(f) + a^*(T(g)f)\) .
Therefore \(F = \pi _s(g)\Omega \)  is a solution of the
following system of equations:
\(a(f)F = - a^*(T(g)f)F \quad \mbox{ for } \quad f \in \mathcal {H}.\) 
Step 2: For each \(n \in {\mathbb {N}}\) , the subset \(a^*(\mathcal {H})S^n(\mathcal {H})\)  is total
in \(S^{n+1}(\mathcal {H})\) , which implies that
\(\lbrace  T \in S^{n+1}(\mathcal {H}) \colon  (\forall f \in \mathcal {H})\, a(f)T = 0\rbrace = \lbrace 0\rbrace .\) 
If an element \(F= \sum _{n = 0}^\infty F_n \in S(\mathcal {H})\) 
with \(F_n \in S^n(\mathcal {H})\)  satisfies
(REF ), then
\(a(f)F_1 = 0, \quad a(f)F_{n+1} = - a^*(T(g)f)F_{n-1} \quad \mbox{ for } \quad f \in \mathcal {H},n \in {\mathbb {N}}.\) 
This implies \(F_1 = 0\) , and inductively we obtain with
(REF ) \(F_{2k+1} = 0\)  for \(k \in {\mathbb {N}}_0\) .
We also derive from (REF ) that \(F\)  is completely determined
by \(F_0\) , hence that the solution space of (REF )
is at most one-dimensional.
If \(F\)  is a solution, we may w.l.o.g. assume that \(F_0 = \Omega \) .
Then \(F_2\)  satisfies
\( a(f) F_2 = - a^*(T(g)f) \Omega = - T(g)f, \quad f \in \mathcal {H}, \) 
i.e., \(F_2 = - \widehat{T(g)}\)  (Lemma REF ).
This observation implies Shale's
result that only for \(g \in \mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})\) , i.e.,
\(\Vert g_2 \Vert _2 < \infty \) , equation
() has a solution \(\pi _s(g)\in \mathop {\rm U{}}\nolimits (S(\mathcal {H}))\) .
Step 3: Combining Lemma REF (i) with
Remark REF , we conclude that the exponential series
\(e^{-\widehat{T(g)}}\)  converges in \(S(\mathcal {H})\) 
for \(\Vert T(g)\Vert _2 < 1\) , which holds on an open \(\mathbf {1}\) -neighborhood in
\(\mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})\) .
Since the operators \(a(f)\)  act as derivations on \(S(\mathcal {H})_0\) , it follows that
\( a(f) e^{-\widehat{T(g)}}= - a(f) \widehat{T(g)} \vee e^{-\widehat{T(g)}}= - T(g)f \vee e^{- \widehat{T(g)}}= - a^*(T(g)f) e^{-\widehat{T(g)}}, \) 
so that \(e^{-\widehat{T(g)}}\)  satisfies (REF ).
We conclude that
\(\pi _s(g) \Omega = c(g) e^{-\widehat{T(g)}}, \quad c(g) \in \times .\) 
Choosing the operators \(\pi _s(g)\) , \(g \in \mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})\) ,
in such a way that
\(c(g) = \langle \Omega _g, \Omega \rangle = \langle \pi _s(g)\Omega , \Omega \rangle > 0,\) 
it follows that
\(c(g) = \Vert e^{-\widehat{T(g)}}\Vert ^{-1}\) 
(cf. [37]}).
Step 4: Since the map
\(g \mapsto T(g) = g_2 g_1^{-1}\) 
is smooth in an identity neighborhood and the map
\( \Big \lbrace  A \in {\mathfrak {p}}_2 \colon  \Vert A\Vert _2 < 1\Big \rbrace  \rightarrow S(\mathcal {H}),\quad A \mapsto e^{\widehat{A}} \) 
is analytic (cf. Remark REF  and
the proof of Lemma REF (v) for \({\mathfrak {p}}_2\) ),
hence in particular smooth, it follows that
\(e^{-\widehat{T(g)}}\)  and hence also \(c(g)\)  are smooth in an identity neighborhood
of \(\mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})\) .
From \(\langle \pi _s(g)^*\Omega , \Omega \rangle = \langle \Omega , \pi _s(g)\Omega \rangle > 0\) 
we further obtain \(\pi _s(g)^* = \pi _s(g^{-1})\) .
This implies that the map
\( (g,h) \mapsto \langle \pi _s(g)\pi _s(h)\Omega ,\Omega \rangle = \langle \pi _s(h)\Omega ,\pi _s(g)^*\Omega \rangle = \langle \pi _s(h)\Omega ,\pi _s(g^{-1})\Omega \rangle \) 
is smooth in an identity neighborhood. Now
Theorem REF  implies that
\(\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\)  is a Lie group and that
\(\Omega \)  is a smooth vector for the corresponding unitary representation,
also denoted \(\pi _s\) .
Since \(\mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})\)  acts smoothly on \(\mathop {{\rm Heis}}\nolimits (\mathcal {H})\) , the
space \(S(\mathcal {H})^\infty \)  of smooth vectors is invariant under
\(\mathop {{\rm Heis}}\nolimits (\mathcal {H})\) , so that the irreducibility of the Fock
representation of \(\mathop {{\rm Heis}}\nolimits (\mathcal {H})\)  on \(S(\mathcal {H})\)  ([37]})
implies the smoothness of \(\pi _s\) .
Step 5: With Remark REF  we can now calculate a
suitable cocycle \(\eta \)  with \(F(g) := \pi _s(g)\Omega =c(g) e^{-\widehat{T(g)}}\)  by
\( \eta (x,y)= 2\mathop {{\rm Im}}\nolimits \langle {\tt d}F(\mathbf {1})x, {\tt d}F(\mathbf {1})y\rangle + i \langle {\tt d}F(\mathbf {1})[x,y],\Omega \rangle .\) 
As \(F(g)\)  only depends on \(T(g) = g_2 g_1^{-1}\) , we have
\(F(gu) = F(g)\)  for \(u \in \mathop {\rm U{}}\nolimits (\mathcal {H})\) , which leads to
\({\tt d}F(\mathbf {1})x = 0\)  for \(x \in {\mathfrak {u}}(\mathcal {H})\) . Hence
\(\eta (x,\cdot ) = 0\)  for \(x \in {\mathfrak {u}}(\mathcal {H})\) .
For \(x \in {\mathfrak {p}}_2\)  we find with
\(T(\exp x) = \cosh (x)\sinh (x)^{-1} = \tanh (x)\) 
the relation \({\tt d}T(\mathbf {1})x = x\) , and hence
\( {\tt d}F(\mathbf {1})x = {\tt d}c(\mathbf {1})(x) \Omega -\widehat{x}. \) 
Since \(c\)  is real-valued, this leads with Lemma REF (ii)
for \(x, y \in {\mathfrak {p}}_2\)  to
\(\eta (x,y)&= 2\mathop {{\rm Im}}\nolimits \langle {\tt d}c(\mathbf {1})(x) \Omega - \widehat{x},{\tt d}c(\mathbf {1})(y) \Omega - \widehat{y} \rangle = 2 \mathop {{\rm Im}}\nolimits \langle \widehat{x}, \widehat{y} \rangle \\&= \mathop {{\rm Im}}\nolimits \mathop {{\rm tr}}\nolimits (xy) = {\textstyle {\frac{1}{2}}}\mathop {{\rm Im}}\nolimits \mathop {{\rm tr}}\nolimits ([x,y]) = \frac{1}{2i} \mathop {{\rm tr}}\nolimits ([x,y])\) 

because the trace of the symmetric operator \(xy+yx\)  is real
and \(\mathop {{\rm tr}}\nolimits ([x,y]) \in i {\mathbb {R}}\) .
\(\Box \)
The content of the preceding theorem is essentially known
(cf. [37]}, [65]}, [66]}),
although all references known to the author only discuss
the metaplectic representation
as a representation of a topological group
and not as a Lie group. In [37]}
and [68]} one finds quite explicit formulas for group
cocycles describing \(\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\)  as a central extension.
Remark 9.4  
The metaplectic representation of
\(G := \widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\)  is not irreducible.
Since the representation of \(\mathop {\rm U{}}\nolimits (\mathcal {H})\)  on each \(S^n(\mathcal {H})\) 
is irreducible (Example REF ), every
\(G\) -invariant subspace is the direct sum of some \(S^n(\mathcal {H})\) ,
\(n \in {\mathbb {N}}_0\) .
The preceding proof immediately shows that
\(\pi (G)\Omega \subseteq S^{\rm even}(\mathcal {H}) := \widehat{\oplus }_{n \in {\mathbb {N}}_0} S^{2n}(\mathcal {H})\) 
and that all projections onto the subspaces \(S^{2n}(\mathcal {H})\) 
are non-zero.
Using the fact that the operators in
\({\tt d}\pi _s({\mathfrak {g}})\)  contain all multiplication operators
with elements \(\widehat{A}\) , \(A \in {\mathfrak {p}}_2\)  (cf. Lemma REF ),
and their adjoints ([37]}),
it easily follows that the representations of \(G\)  on the two
subspaces
\(S^{\rm even}(\mathcal {H})\)  and \(S^{\rm odd}(\mathcal {H})\)  are irreducible.

The metaplectic group
In this subsection we describe a convenient description
of the metaplectic group \(\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\)  as a Banach–Lie group.
For further details we refer to [8]}.
First we recall from [8]}
that
\(\mathop {{\rm Sp}}\nolimits _{1,2}(\mathcal {H})&:= \Big \lbrace  g = \begin{pmatrix}  a & b \\ c & d \end{pmatrix} \in \mathop {{\rm Sp}}\nolimits (\mathcal {H}) \subseteq \mathop {\rm U{}}\nolimits (\mathcal {H},\mathcal {H}) \colon  \\&\qquad \qquad \qquad \Vert b\Vert _2, \Vert c\Vert _2 < \infty ,\Vert a - \mathbf {1}\Vert _1, \Vert d - \mathbf {1}\Vert _1 < \infty \Big \rbrace \) 

carries the structure of a Banach–Lie group
with polar decomposition
\( \mathop {\rm U{}}\nolimits _1(\mathcal {H}) \times {\mathfrak {p}}_2 \rightarrow \mathop {{\rm Sp}}\nolimits _{1,2}(\mathcal {H}), \quad {\mathfrak {p}}_2 = \Big \lbrace  \begin{pmatrix}  0 & b \\ b^* & 0 \end{pmatrix} \colon  b = b^\top \in B_2(\mathcal {H})\Big \rbrace \) 
(cf. [8]}).
The full unitary group \(\mathop {\rm U{}}\nolimits (\mathcal {H})\)  acts smoothly by conjugation
on \(\mathop {{\rm Sp}}\nolimits _{1,2}(\mathcal {H})\) , so that we can form the semidirect product
\(\mathop {{\rm Sp}}\nolimits _{1,2}(\mathcal {H}) \rtimes \mathop {\rm U{}}\nolimits (\mathcal {H})\) , and the multiplication map
\( \mu _1 \colon  \mathop {{\rm Sp}}\nolimits _{1,2}(\mathcal {H}) \rtimes \mathop {\rm U{}}\nolimits (\mathcal {H}) \rightarrow \mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H}),\quad (g,u) \mapsto gu \) 
is a quotient morphism of Banach–Lie groups with kernel
\( N := \lbrace  (g,g^{-1}) \colon  g \in \mathop {\rm U{}}\nolimits _1(\mathcal {H}) \rbrace , \) 
so that
\( \mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H}) \cong (\mathop {{\rm Sp}}\nolimits _{1,2}(\mathcal {H}) \rtimes \mathop {\rm U{}}\nolimits (\mathcal {H}))/N \) 
(cf. [8]}).
From the polar decomposition of
\(\mathop {{\rm Sp}}\nolimits _{1,2}(\mathcal {H})\)  we derive that
\(\pi _1(\mathop {{\rm Sp}}\nolimits _{1,2}(\mathcal {H})) \cong \pi _1(\mathop {\rm U{}}\nolimits _1(\mathcal {H})) \cong {\mathbb {Z}}\) , hence the
existence of a unique 2-fold covering group
\(q \colon  \mathop {\rm Mp{}}\nolimits _{1,2}(\mathcal {H})\rightarrow \mathop {{\rm Sp}}\nolimits _{1,2}(\mathcal {H})\) . On the inverse
image \(\widehat{\mathop {\rm U{}}\nolimits }_1(\mathcal {H})\)  of \(\mathop {\rm U{}}\nolimits _1(\mathcal {H})\)  in \(\mathop {{\rm Sp}}\nolimits _{1,2}(\mathcal {H})\)  we then
have a unique character
\(\sqrt{\det } \colon  \widehat{\mathop {\rm U{}}\nolimits }_1(\mathcal {H}) \rightarrow \mbox{ with} \quad \mathop {\bf L{}}\nolimits (\sqrt{\det }) = \frac{1}{2i} \mathop {{\rm tr}}\nolimits .\) 
Next we observe that the smooth action of \(\mathop {\rm U{}}\nolimits (\mathcal {H})\)  on
\(\mathop {{\rm Sp}}\nolimits _{1,2}(\mathcal {H})\) 
lifts to a smooth action on \(\mathop {\rm Mp{}}\nolimits _{1,2}(\mathcal {H})\) . We also note that,
for \(\mathop {{\rm SU}}\nolimits _1(\mathcal {H}) := \ker (\det )\) , we have
\(\mathop {\rm U{}}\nolimits _1(\mathcal {H}) \cong \mathop {{\rm SU}}\nolimits _1(\mathcal {H}) \rtimes , where the determinantis the projection onto the second factor. Accordingly,\) U1(H) SU1(H)  with
\(q(g,t) = (g,t^2)\)  and \(\sqrt{\det }(g,t) = t\) .
Writing \(x = x_1 + x_2\)  for the decomposition of \(x \in \mathfrak {sp}(\mathcal {H})\) 
into linear and antilinear component,
we recall from Theorem REF  that
\( \widehat{\mathfrak {sp}}_{\rm res}(\mathcal {H}) \cong {\mathbb {R}}\oplus _\eta \mathfrak {sp}_{\rm res}(\mathcal {H})\quad \mbox{ with } \quad \eta (x,y) = \frac{1}{2i} \mathop {{\rm tr}}\nolimits ([x_2, y_2]). \) 
This implies that
\( \sigma \colon  \mathfrak {sp}_{1,2}(\mathcal {H}) \rightarrow \widehat{\mathfrak {sp}}_{\rm res}(\mathcal {H})\cong {\mathbb {R}}\oplus _\eta \mathfrak {sp}_{\rm res}(\mathcal {H}), \quad \sigma (x) := \Big (\frac{1}{2i} \mathop {{\rm tr}}\nolimits (x_1), x\Big ) \) 
is a homomorphism of Banach–Lie algebras. Here we use that
\( [x,y]_1 = [x_1, y_1] + [x_2, y_2]\quad \mbox{ and } \quad \mathop {{\rm tr}}\nolimits ([x_1, y_1])=0. \) 
On the subgroup \(\widehat{\mathop {\rm U{}}\nolimits }_1(\mathcal {H}) \subseteq \mathop {\rm Mp{}}\nolimits _{1,2}(\mathcal {H})\) ,
\(\sigma \)  integrates to the group homomorphism
\(\sigma _G \colon  \widehat{\mathop {\rm U{}}\nolimits }_1(\mathcal {H}) \rightarrow \widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\subseteq \mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H}) \times \mathop {\rm U{}}\nolimits (S(\mathcal {H})), \quad \sigma (g) := (q(g), \sqrt{\det }(g)\pi _s(g)), \) 
so that the
polar decomposition of \(\mathop {{\rm Sp}}\nolimits _{1,2}(\mathcal {H})\)  implies that
\(\sigma \)  integrates to a morphism of Banach–Lie groups
\(\sigma _G \colon  \mathop {\rm Mp{}}\nolimits _{1,2}(\mathcal {H}) \rightarrow \widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H}).\) 
Combining this map with the canonical inclusion
\(\mathop {\rm U{}}\nolimits (\mathcal {H}) \hookrightarrow \widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\) , the equivariance of
\(\sigma _G\)  under conjugation with unitary operators implies the
existence of a homomorphism
\( \mu \colon  \mathop {\rm Mp{}}\nolimits _{1,2}(\mathcal {H}) \rtimes \mathop {\rm U{}}\nolimits (\mathcal {H}) \rightarrow \widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H}),\quad (g,u) \mapsto \sigma _G(g)u.\) 
Proposition 9.5   The homomorphism \(\mu \) 
factors through an isomorphism
\( \overline{\mu }\colon  (\mathop {\rm Mp{}}\nolimits _{1,2}(\mathcal {H}) \rtimes \mathop {\rm U{}}\nolimits (\mathcal {H}))/\ker \mu \rightarrow \widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\) 
of connected Banach–Lie groups with
\(\ker \mu \cong \mathop {{\rm SU}}\nolimits _1(\mathcal {H}).\)

Proof.   
If \(\mu (g,u) = \mathbf {1}\) , then \(\sigma _G(g) = u^{-1}\)  implies that
\(g \in \widehat{\mathop {\rm U{}}\nolimits }_1(\mathcal {H})\)  with \(q(g) = u^{-1}\) .
Now
\( \Omega = \pi _s(\mu (g,u)) \Omega = \sqrt{\det }(g)\Omega \) 
implies \(\sqrt{\det }(g) = 1\) . This shows that
\(\ker \mu = \lbrace (g,q(g)^{-1}) \colon  g \in \mathop {{\rm SU}}\nolimits _1(\mathcal {H})\rbrace ,\) 
and the assertion follows.
\(\Box \)
The inverse image of
the center \(Z \cong  of \) Spres(H)\( is\)\(\widehat{N} := \lbrace  (g,q(g)^{-1}) \colon  g \in \widehat{\mathop {\rm U{}}\nolimits }_1(\mathcal {H})\rbrace .\)\(The inverse image \) U(H)\( of the subgroup \) U(H)\( in\) Spres(H)\( is of the form\begin{equation}\widehat{\mathop {\rm U{}}\nolimits }(\mathcal {H}) \cong (\widehat{\mathop {\rm U{}}\nolimits }_1(\mathcal {H}) \rtimes \mathop {\rm U{}}\nolimits (\mathcal {H}))/\ker \mu \cong Z \times \mathop {\rm U{}}\nolimits (\mathcal {H}).\end{equation}In particular, it splits as a central extension of Lie groups.\)
Semiboundedness of the metaplectic representation
In this subsection we show that the metaplectic representation
\((\pi _s, S(\mathcal {H}))\)  of \(\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\)  is semibounded and
determine the open invariant cone \(W_{\pi _s}\)  as the inverse
image of the canonical cone \(W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}\)  in
\(\mathfrak {sp}_{\rm res}(\mathcal {H})\) .
We start with a closer look at adjoint orbits of the
symplectic Lie algebra.
Lemma 9.6   If \((V,\omega )\)  is a finite dimensional
symplectic space and \(X \in \mathfrak {sp}(V,\omega )\)  is such that
the Hamiltonian function \(H_X(v) = {\textstyle {\frac{1}{2}}}\omega (Xv,v)\)  is indefinite, then
\(\mathop {{\rm conv}}\nolimits (\mathcal {O}_X) = \mathfrak {sp}(V,\omega )\) .

Proof.   
Let \({\mathfrak {g}}:= \mathfrak {sp}(V,\omega )\) . Then Proposition REF ,
combined with the uniqueness of the open invariant cone in
\(\mathfrak {sp}(V,\omega )\)  up to sign
(cf. [17]}),
implies that the set of semi-equicontinuous coadjoint
orbits coincides with the double cone
\(W_{\mathfrak {sp}(V,\omega )}^\star \cup - W_{\mathfrak {sp}(V,\omega )}^\star .\)
Using the Cartan–Killing form to identify \({\mathfrak {g}}\)  with its dual,
we accordingly see that any semi-equicontinuous
adjoint orbit is contained in the double cone
\( \overline{W_{\mathfrak {sp}(V,\omega )}} \cup - \overline{W_{\mathfrak {sp}(V,\omega )}}, \) 
i.e., the corresponding Hamiltonian function is either
positive or negative.
Finally, we observe that if \(x \in {\mathfrak {g}}\)  satisfies
\(\mathop {{\rm conv}}\nolimits (\mathcal {O}_x) \ne {\mathfrak {g}}\) , then \(B(\mathcal {O}_x)\)  is a non-zero invariant
cone and Proposition REF (i)
implies that it has interior points.
Therefore \(\mathcal {O}_x\)  is semi-equicontinuous because every finite dimensional
space is barrelled.
\(\Box \)
Lemma 9.7   The subspace
\({\mathfrak {p}}:= \lbrace  X \in \mathfrak {sp}(\mathcal {H}) \colon  IX = - XI\rbrace \) 
contains no non-trivial Lie algebra ideals, and the same holds
for the subspace
\({\mathfrak {p}}_2 := \lbrace  X \in \mathfrak {sp}_{\rm res}(\mathcal {H}) \colon  IX = - XI\rbrace = {\mathfrak {p}}\cap \mathfrak {sp}_{\rm res}(\mathcal {H}).\) 


Proof.   
First we show that \(X \in {\mathfrak {p}}\)  and \([X,[I,X]] = 0\)  implies
\(X = 0\) . Realizing \(\mathfrak {sp}(\mathcal {H})\)  as a closed subalgebra of
\(\mathop {{\mathfrak {gl} }}\nolimits (\mathcal {H}_ \cong \mathop {{\mathfrak {gl} }}\nolimits (\mathcal {H}\oplus \mathcal {H})\)  (cf. Example REF (h)),
we have
\( {\mathfrak {p}}= \Big \lbrace  \begin{pmatrix}  0 & a \\ a^* & 0 \end{pmatrix} \colon  a^\top = a \in B(\mathcal {H})\Big \rbrace \quad \mbox{ and } \quad I = \begin{pmatrix}  i & 0 \\ 0 & -i \end{pmatrix}. \) 
For \(X = \begin{pmatrix}  0 & a \\ a^* & 0 \end{pmatrix}\)  this leads to
\([X,[I,X]]&= \Big [\begin{pmatrix}  0 & a \\ a^* & 0 \end{pmatrix}, \begin{pmatrix}  0 & 2ia \\ -2ia^* & 0 \end{pmatrix}\Big ]= -4i\begin{pmatrix}  aa^* & 0 \\ 0& a^* a \end{pmatrix}.\) 

If this operator vanishes, then \(a^* a = 0\)  implies that
\(a = 0\) , so that \(X = 0\) .
If \({\mathfrak {i}}\subseteq {\mathfrak {p}}\)  is a Lie algebra ideal,
then we have for each \(X \in {\mathfrak {i}}\)  the relation
\([X,[I,X]] \in {\mathfrak {i}}\cap {\mathfrak {u}}(\mathcal {H}) = \lbrace 0\rbrace \) , so that
the preceding argument shows that \(X = 0\) . The same argument
applies to \({\mathfrak {p}}_2 = {\mathfrak {p}}\cap \mathfrak {sp}_{\rm res}(\mathcal {H})\) .
\(\Box \)
Lemma 9.8   Let \(X \in \mathfrak {sp}(\mathcal {H})\) . Then
the projection
\(p_{\mathfrak {k}}\colon  \mathfrak {sp}(\mathcal {H}) \rightarrow {\mathfrak {u}}(\mathcal {H})\) 
onto the \(-linear component satisfies\)\( p_{\mathfrak {k}}(\mathcal {O}_X) \subseteq \lbrace  y \in {\mathfrak {u}}(\mathcal {H}) \colon  i y \le 0 \rbrace = \overline{C_{{\mathfrak {u}}(\mathcal {H})}}, \)\(if and only if \) HX 0\(. A correspondingstatement holds for \) spres(H)\(.\)
Proof.   
Since \(p_{{\mathfrak {k}}}\)  is the fixed point projection for the
action of the torus \(e^{{\mathbb {R}}\mathop {{\rm ad}}\nolimits I}\) , it preserves the closed convex cones
\(\pm \lbrace  Z \in \mathfrak {sp}(\mathcal {H}) \colon  H_Z \ge 0\rbrace \)  (Proposition REF ),
so that the relation
\(H_X \ge 0\)  implies \(i p_{\mathfrak {k}}(\mathcal {O}_X) \le 0\) 
(Theorem REF (i)).
Suppose, conversely, that \(i p_{\mathfrak {k}}(\mathcal {O}_X) \le 0\) .
If \(H_X \le 0\) , then the preceding argument
shows that \(i p_{\mathfrak {k}}(\mathcal {O}_X) \ge 0\) , which leads to
\(p_{\mathfrak {k}}(\mathcal {O}_X) = \lbrace 0\rbrace \) , so that the closed span of \(\mathcal {O}_X\)  is
an ideal of \({\mathfrak {g}}\)  contained in \({\mathfrak {p}}= \ker p_{\mathfrak {k}}\) .
In view of Lemma REF , this leads to \(X = 0\) .
We may therefore assume that \(H_X\)  is indefinite.
Hence there exists a finite dimensional
complex subspace \(\mathcal {H}_1 \subseteq \mathcal {H}\)  on which \(H_X\)  is indefinite.
Then we have a Hilbert space direct sum
\(\mathcal {H}= \mathcal {H}_1 \oplus \mathcal {H}_1^\bot \)  and accordingly we write
operators on \(\mathcal {H}\)  as \((2 \times 2)\) -block matrices:
\( X = \begin{pmatrix}  X_{11} & X_{12}\\ X_{21} & X_{22} \end{pmatrix}. \) 
For \(v \in \mathcal {H}_1\)  we then have
\(\mathop {{\rm Im}}\nolimits \langle Xv,v \rangle = \mathop {{\rm Im}}\nolimits \langle X_{11}v,v \rangle ,\) 
so that \(X_{11} \in \mathfrak {sp}(\mathcal {H}_1)\)  and \(H_{X_{11}}\)  is indefinite.
Identifying \(\mathop {{\rm Sp}}\nolimits (\mathcal {H}_1)\)  in the natural way with a subgroup
of \(\mathop {{\rm Sp}}\nolimits (\mathcal {H})\) , Lemma REF  implies that
\(\mathop {{\rm conv}}\nolimits (\mathop {{\rm Ad}}\nolimits (\mathop {{\rm Sp}}\nolimits (\mathcal {H}_1)X))\)  contains an element
\(Y\)  with \(Y_{11} = - i \mathbf {1}\) .
From
\( p_{\mathfrak {k}}(Y)= \begin{pmatrix}  p_{\mathfrak {k}}(Y_{11}) & p_{\mathfrak {k}}(Y_{12})\\p_{\mathfrak {k}}(Y_{21}) & p_{\mathfrak {k}}(Y_{22}) \end{pmatrix}= \begin{pmatrix}  Y_{11} & p_{\mathfrak {k}}(Y_{12})\\p_{\mathfrak {k}}(Y_{21}) & p_{\mathfrak {k}}(Y_{22}) \end{pmatrix} \) 
we now derive that \(i p_{\mathfrak {k}}(Y) \lnot \le 0\) , contradicting
our assumption on \(X\) . This completes the proof of the
first assertion.
Since \(\mathop {{\rm Sp}}\nolimits (\mathcal {H}_1) \subseteq \mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})\) , the preceding
argument also implies the assertion for the restricted
Lie algebra \(\mathfrak {sp}_{\rm res}(\mathcal {H})\) .
\(\Box \)
Now we are ready to show that the metaplectic representation
is semibounded. We write
\(q \colon  \widehat{\mathfrak {sp}}_{\rm res}(\mathcal {H}) \rightarrow \mathfrak {sp}_{\rm res}(\mathcal {H})\) 
for the quotient map and \({\mathfrak {z}}\)  for its kernel.
Since the central extension is trivial over the subalgebra
\({\mathfrak {u}}(\mathcal {H})\)  of \(\mathfrak {sp}_{\rm res}(\mathcal {H})\) , we have
\( \widehat{{\mathfrak {u}}}(\mathcal {H}) := q^{-1}({\mathfrak {u}}(\mathcal {H})) \cong {\mathfrak {z}}\oplus {\mathfrak {u}}(\mathcal {H}), \) 
where the \({\mathfrak {u}}(\mathcal {H})\) -complement is uniquely determined by the
property that it is the commutator algebra.
Here we use that the Lie algebra \({\mathfrak {u}}(\mathcal {H})\)  is perfect
(cf. [8]}).
Accordingly, we may identify \({\mathfrak {u}}(\mathcal {H})\)  in a natural
way with a subalgebra of \(\widehat{\mathfrak {sp}}_{\rm res}(\mathcal {H})\) .
Remark 9.9   The momentum set \(I_{\pi _s}\) 
is completely
determined by the restriction of its support function
\(s_{\pi _s}\)  to \(W_{\pi _s} = B(I_{\pi _s})^0\)  (Remark REF (b))
which is invariant under the adjoint action.
We shall see below that \(q(W_{\pi _s}) \subseteq W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}\) ,
so that Theorem REF (ii) leads to
\( W_{\pi _s} \subseteq \mathop {{\rm Ad}}\nolimits (\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H}))({\mathfrak {z}}\times C_{{\mathfrak {u}}(\mathcal {H})}), \) 
where \({\mathfrak {z}}\cong {\mathbb {R}}\)  denotes the center. This
in turn entails that \(I_{\pi _s}\)  is already determined
by the restriction of \(s_{\pi _s}\)  to \({\mathfrak {z}}\times C_{{\mathfrak {u}}(\mathcal {H})}\) ,
which we determine below.
This restriction is the support
function for the momentum set of the restriction of
\({\pi _s}\)  to the subgroup \(\widehat{\mathop {\rm U{}}\nolimits }(\mathcal {H}) \cong \mathop {\rm U{}}\nolimits (\mathcal {H})\) 
(Proposition REF (iii)), which is a
direct sum of the representations on the subspaces \(S^n(\mathcal {H})\) , on
which we have
\( {\pi _s}(z,g)(v_1 \vee \cdots \vee v_n)= z gv_1 \vee \cdots \vee g v_n \quad \mbox{ for } \quad v_1,\ldots , v_n \in \mathcal {H}. \) 
For the representation of \(\mathop {\rm U{}}\nolimits (\mathcal {H})\)  on \(S^n(\mathcal {H})\) , the momentum
set \(I_{S^n(\mathcal {H})}^{\mathop {\rm U{}}\nolimits (\mathcal {H})}\)  is simply given by
\( I_{S^n(\mathcal {H})}^{\mathop {\rm U{}}\nolimits (\mathcal {H})} = n I_{\mathcal {H}}^{\mathop {\rm U{}}\nolimits (\mathcal {H})} \) 
(Example REF (b)), so that
\( I_{\pi _s}^{\widehat{U}(\mathcal {H})}= \overline{\mathop {{\rm conv}}\nolimits }\big (\bigcup _{n \in {\mathbb {N}}_0} \lbrace 1\rbrace  \times n I_\mathcal {H}^{\mathop {\rm U{}}\nolimits (\mathcal {H})}\big ), \) 
which is a convex cone with vertex \((1,0)\) , and for
\(x \in {\mathfrak {u}}(\mathcal {H})\) , we have
\(s_{\pi _s}(t,x) ={\left\lbrace \begin{array}{ll}-t & \text{ for } ix \le 0 \\\infty & \text{ else}. \\\end{array}\right.}\) 
The representation of \(\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\)  on
\(S(\mathcal {H})\)  decomposes into two irreducible pieces
\( S^{\rm even}(\mathcal {H}) = \widehat{\oplus }_{n \in {\mathbb {N}}_0} S^{2n}(\mathcal {H})\quad \mbox{ and } \quad S^{\rm odd}(\mathcal {H}) = \widehat{\oplus }_{n \in {\mathbb {N}}_0} S^{2n+1}(\mathcal {H}) \) 
(Remark REF ).
Writing \(\pi _\pm \)  for the corresponding representations, we obtain
\( I_{\pi _+}^{\widehat{U}(\mathcal {H})} = I_{{\pi _s}}^{\widehat{U}(\mathcal {H})}= \overline{\mathop {{\rm conv}}\nolimits }\Big (\bigcup _{n \in {\mathbb {N}}_0} \lbrace 1\rbrace  \times 2n I_\mathcal {H}^{\mathop {\rm U{}}\nolimits (\mathcal {H})}\Big ), \) 
and
\( I_{\pi _-}^{\widehat{U}(\mathcal {H})}= \overline{\mathop {{\rm conv}}\nolimits }\Big (\bigcup _{n \in {\mathbb {N}}_0} \lbrace 1\rbrace  \times (2n+1)I_\mathcal {H}^{\mathop {\rm U{}}\nolimits (\mathcal {H})}\Big ). \) 
Now it is easy to derive that
\(W_{\pi _\pm } = W_{\pi _s}\) , \(s_{\pi _+} = s_{\pi _s}\) , and that
\( s_{\pi _-}(t,x) = -t + \sup ({\rm Spec}(ix)) \quad \mbox{ for } \quad ix \le 0. \) 

Theorem 9.10  
The metaplectic representation
\(({\pi _s}, S(\mathcal {H}))\)  of
\(\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H}))\)  is semibounded with
\( W_{\pi _s} = W_{\widehat{\mathfrak {sp}}_{\rm res}(\mathcal {H})} := q^{-1}(W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}). \) 


Proof.   
From Example REF  we know that
\( s_{\pi _s}(x) \le 0 \quad \mbox{ for } \quad x \in C_{{\mathfrak {u}}(\mathcal {H})},\quad \mbox{ so that } \quad s_{\pi _s}(t,x) = -t + s_{\pi _s}(x) < \infty \) 
for \((t,x) \in {\mathfrak {z}}\times C_{{\mathfrak {u}}(\mathcal {H})}\) .
From the invariance of \(s_{\pi _s}\)  under the adjoint action
and Theorem REF (ii) it now follows that
\(\mathop {{\rm Ad}}\nolimits (\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H}))({\mathfrak {z}}\times C_{{\mathfrak {u}}(\mathcal {H})})= q^{-1}(\mathop {{\rm Ad}}\nolimits (\mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H}))C_{{\mathfrak {u}}(\mathcal {H})})= q^{-1}(W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}) \subseteq B(I_{\pi _s})\) 
(cf. Theorem REF (ii)).
In particular, \(B(I_{\pi _s})\)  has interior points,
so that \({\pi _s}\)  is semibounded with \(W_{\widehat{\mathfrak {sp}}_{\rm res}(\mathcal {H})} \subseteq W_{\pi _s}\) .
To prove equality, we note that (REF ) in
Remark REF  implies that
\( W_{\pi _s} \cap {\mathfrak {u}}(\mathcal {H}) \subseteq \lbrace  x \in {\mathfrak {u}}(\mathcal {H}) \colon  s_{\pi _s}(x) < \infty \rbrace = \lbrace  x \in {\mathfrak {u}}(\mathcal {H}) \colon  ix \le 0\rbrace  = \overline{C_{{\mathfrak {u}}(\mathcal {H})}}. \) 
As \(W_{\pi _s}\)  is invariant under the projection
\(p_{\mathfrak {k}}\colon  \widehat{\mathfrak {sp}}_{\rm res}(\mathcal {H}) \rightarrow \widehat{{\mathfrak {u}}}(\mathcal {H})\) 
onto the fixed point space of the compact group
\(e^{{\mathbb {R}}\mathop {{\rm ad}}\nolimits I}\)  (Proposition REF ),
we obtain for each \(x \in W_{\pi _s}\)  the relation
\(i p_{\mathfrak {k}}(\mathcal {O}_{q(x)})\le 0,\) 
so that Lemma REF  leads to
\(q(x) \in \overline{W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}}\) . As \(W_{\pi _s}\)  is open and
\(W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}\)  coincides with the interior of its closure
(Lemma REF ), it follows that
\( q(W_{\pi _s}) \subseteq W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}. \) 
Combining this with (REF ),
we obtain \(q(W_{\pi _s}) = W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}.\)
As the center \({\mathfrak {z}}\cong {\mathbb {R}}\)  of \(\widehat{\mathfrak {sp}}_{\rm res}(\mathcal {H})\) 
acts by multiples of the identity on \(S(\mathcal {H})\) ,
we have \({\mathfrak {z}}\subseteq H(W_{\pi _s})\) , which finally leads to
\(W_{\pi _s} = q^{-1}(q(W_{\pi _s})) = W_{\widehat{\mathfrak {sp}}_{\rm res}(\mathcal {H})}.\) 
\(\Box \)
The momentum set of the metaplectic representation
Now that we have determined the cone \(W_{\pi _s}\)  for the metaplectic
representation, we now apply the tools from Section 
to determine the momentum set for the representation of the
central extension \(\widehat{\mathop {\rm HSp{}}\nolimits }_{\rm res}(\mathcal {H})\)  acting on
\(S(\mathcal {H})\) . In particular, we show that the momentum set is the
closed convex hull of a single coadjoint orbit.
Definition 9.11  We define
\(\widehat{\mathop {\rm HSp{}}\nolimits }_{\rm res}(\mathcal {H})\) 

as the quotient of the semidirect product
\( \mathop {{\rm Heis}}\nolimits (\mathcal {H}) \rtimes _\alpha \widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H}), \quad \alpha (g,u)((t,v)) := (t,gv), \) 
by the central subgroup
\( S := \lbrace  (t,(\mathbf {1},z\mathbf {1})) \in {\mathbb {R}}\times Z \colon  e^{it}z = 1\rbrace  \cong {\mathbb {R}}\) 
which acts trivially on \(S(\mathcal {H})\) .
This means that \(\widehat{\mathop {\rm HSp{}}\nolimits }_{\rm res}(\mathcal {H})\)  is a central extension
of \(\mathcal {H}\rtimes \mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})\)  by \(. The correspondingLie algebra cocycle is given by\)\(((v,x), (v^{\prime },x^{\prime })) \mapsto \mathop {{\rm Im}}\nolimits \langle v,v^{\prime }\rangle + \eta (x,x^{\prime })= \mathop {{\rm Im}}\nolimits \langle v,v^{\prime }\rangle + {\textstyle {\frac{1}{2}}}\mathop {{\rm Im}}\nolimits \mathop {{\rm tr}}\nolimits (x_2 x^{\prime }_2) \)\((cf.\ Theorem~\ref {thm:metasmooth}).\)
Remark 9.12 
The representations \(W\)  of \(\mathop {{\rm Heis}}\nolimits (\mathcal {H})\)  and \(\pi _s\)  of
\(\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\)  combine to a representation, also denoted
\(\pi _s\)  of
\(\widehat{\mathop {\rm HSp{}}\nolimits }_{\rm res}(\mathcal {H})\)  on \(S(\mathcal {H})\) . Since \((W,S(\mathcal {H}))\)  is irreducible,
the extension to \(\widehat{\mathop {\rm HSp{}}\nolimits }_{\rm res}(\mathcal {H})\)  is also irreducible.
To see that it is smooth, it suffices to show that the function
\( \mathop {{\rm Heis}}\nolimits (\mathcal {H}) \rtimes \widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\rightarrow  \quad (h,g) \mapsto \langle W(h) \pi _s(g)\Omega , \Omega \rangle = \langle \pi _s(g)\Omega , W(h)^{-1}\Omega \rangle \) 
is smooth (Theorem REF ).
This follows from the smoothness
of the vector \(\Omega \)  for \(\mathop {{\rm Heis}}\nolimits (\mathcal {H})\)  and \(\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\) 
(cf. (REF ) and Theorem REF ).

Proposition 9.13  The metaplectic representation \((\pi _s, S(\mathcal {H}))\) 
of \(\widehat{\mathop {\rm HSp{}}\nolimits }_{\rm res}(\mathcal {H})\)  is semibounded with
\( W_{\pi _s} = q^{-1}(\mathcal {H}\times W_{\mathfrak {sp}_{\rm res}(\mathcal {H})})= \mathop {{\rm Ad}}\nolimits (\widehat{\mathop {\rm HSp{}}\nolimits }_{\rm res}(\mathcal {H}))({\mathfrak {z}}\times C_{{\mathfrak {u}}(\mathcal {H})}), \) 
where \(q \colon  \widehat{\mathfrak {hsp}}_{\rm res}(\mathcal {H}) \rightarrow \mathcal {H}\rtimes \mathfrak {sp}_{\rm res}(\mathcal {H})\) 
is the quotient map whose kernel is the center \({\mathfrak {z}}\cong {\mathbb {R}}\) .

Proof.   
Let
\(p \colon  \mathfrak {hsp}_{\rm res}(\mathcal {H}) = \mathfrak {heis}(\mathcal {H}) \rtimes \mathfrak {sp}_{\rm res}(\mathcal {H})\rightarrow \mathcal {H}\rtimes \mathfrak {sp}_{\rm res}(\mathcal {H}), \quad (t,v,x) \mapsto (v,x) \) 
denote the quotient map, so that Theorem REF (iv) implies that
\( W_{\mathfrak {hsp}_{\rm res}(\mathcal {H})} = p^{-1}(W_{\mathfrak {sp}_{\rm res}(\mathcal {H})})= \mathop {{\rm Ad}}\nolimits (\mathop {\rm HSp{}}\nolimits _{\rm res}(\mathcal {H})) W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}. \) 
Modulo the center, this relation leads to
\(\mathcal {H}\rtimes W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}= \mathop {{\rm Ad}}\nolimits (\mathcal {H}\rtimes \mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H}))W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}= \mathop {{\rm Ad}}\nolimits (\mathcal {H}\rtimes \mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H}))C_{{\mathfrak {u}}(\mathcal {H})}.\) 
From Theorem REF  we know that
\(s_{\pi _s}(x)\)  is finite if \(x \in {\mathfrak {z}}\times C_{{\mathfrak {u}}(\mathcal {H})}\) ,
so that (REF ) implies that
it is also finite if \(q(x) \in \mathcal {H}\rtimes W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}\) , i.e.,
\( W_{\pi _s} \supseteq q^{-1}(\mathcal {H}\rtimes W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}). \) 
This proves already that \(\pi _s\)  is semibounded
because \(W_{\pi _s}\)  has interior points (Proposition REF ).
We further conclude that \(\mathfrak {heis}(\mathcal {H}) \subseteq H(W_{\pi _s})\) , so that
we obtain with Theorem REF  that
\(W_{\pi _s}&= \mathfrak {heis}(\mathcal {H}) + (W_{\pi _s} \cap \widehat{\mathfrak {sp}}_{\rm res}(\mathcal {H}))\subseteq \mathfrak {heis}(\mathcal {H}) + q^{-1}(W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}) \\&= q^{-1}(\mathcal {H}\times W_{\mathfrak {sp}_{\rm res}(\mathcal {H})}).\) 

This proves the desired equality.
Finally, we note that (REF ) implies that every element
\(x \in W_{\pi _s}\)  is conjugate to an element \(y \in q^{-1}(C_{{\mathfrak {u}}(\mathcal {H})})\) ,
which means that \(y \in {\mathfrak {z}}\times C_{{\mathfrak {u}}(\mathcal {H})}\) .
\(\Box \)
Theorem 9.14  
The momentum set \(I_{\pi _s}\)  of the metaplectic representation
of \(\widehat{\mathop {\rm HSp{}}\nolimits }_{\rm res}(\mathcal {H})\)  is the closed convex hull of the coadjoint
orbit of \(\lambda := \Phi _{\pi _s}([\Omega ])\) , and this linear functional
is given by
\(\lambda (t,x) = t\quad \mbox{ on } \quad \widehat{\mathfrak {hsp}}_{\rm res}(\mathcal {H}) = {\mathbb {R}}\oplus (\mathcal {H}\rtimes \mathfrak {sp}_{\rm res}(\mathcal {H})).\) 


Proof.   
Since the metaplectic representation of
\(G := \widehat{\mathop {\rm HSp{}}\nolimits }_{\rm res}(\mathcal {H})\)  is irreducible,
we want to apply Theorem REF .
For the subgroup \(K := \widehat{\mathop {\rm U{}}\nolimits }(\mathcal {H})\subseteq G\)  with Lie algebra
\({\mathfrak {k}}= {\mathfrak {z}}\oplus {\mathfrak {u}}(\mathcal {H})\) , we have
\( {\mathfrak {g}}_{\mathfrak {p}}_+ \oplus {\mathfrak {k}}_{\mathfrak {p}}_-,\quad \mbox{ where } \quad {\mathfrak {p}}_\pm = \ker (\mathop {{\rm ad}}\nolimits I \mp i\mathbf {1})\oplus \ker (\mathop {{\rm ad}}\nolimits I \mp 2i\mathbf {1}), \) 
\(I \in {\mathfrak {u}}(\mathcal {H})\)  is the multiplication with \(i\)  on \(\mathcal {H}\) ,
\( \mathfrak {heis}(\mathcal {H})_ {\mathfrak {z}}_\mathcal {H}_+ \oplus \mathcal {H}_-, \quad \mathcal {H}_\pm := \ker (\mathop {{\rm ad}}\nolimits I \mp i\mathbf {1})\subseteq \mathcal {H}_ \) 
Since the Lie algebra \({\mathfrak {k}}\)  of \(K\)  is complemented by the closed
subspace \(\mathcal {H}\oplus {\mathfrak {p}}_2\)  in \({\mathfrak {g}}\) , the coset space
\(G/K\)  carries the structure of a Banach homogeneous space.
Moreover, the closed \(K\) -invariant subalgebra
\({\mathfrak {k}}_{\mathfrak {p}}_+\)  determines on \(G/K\)  the structure of a
complex manifold for which the tangent space in the base point
can be identified with \({\mathfrak {g}}_({\mathfrak {k}}_ {\mathfrak {p}}_+) \cong {\mathfrak {p}}_-\) 
([76]}).
Let \(\Omega \in S^0(\mathcal {H}) \subseteq S(\mathcal {H})\)  be the vacuum vector.
Then \(\Omega \)  is a smooth vector (Remark REF ),
which is an eigenvector for \(K\) . On \(S(\mathcal {H})\) , the operator
\(-i{\tt d}\pi (I)\)  is diagonal, and \(S^n(\mathcal {H})\)  is the eigenspace corresponding
to the eigenvalue \(n\) . As \(-i\mathop {{\rm ad}}\nolimits I\)  acts on \({\mathfrak {p}}_-\)  with the
eigenvalues \(-1\)  and \(-2\) , it follows that
\({\tt d}\pi _s({\mathfrak {p}}_-)\Omega = \lbrace 0\rbrace ,\) 
so that Theorem REF  provides a holomorphic equivariant map
\( \eta \colon  G/K\rightarrow {\mathbb {P}}(S(\mathcal {H})^{\prime }), \quad g \mapsto \langle \cdot , \pi _s(g)^{-1} \Omega \rangle \) 
and a realization of \((\pi _s, S(\mathcal {H}))\)  by holomorphic sections of a
complex line bundle over \(G/K\) . Since the representation
\((\pi _s, S(\mathcal {H}))\)  of \(G\)  is irreducible, \(\Omega \)  is a cyclic vector.
It remains to show that for each \(x \in W_\pi \)  the flow generated by
\(-x\)  on \(G/K\)  extends holomorphically to \(+\) . Since every such
element \(x\)  is conjugate to an element of \({\mathfrak {z}}\times C_{{\mathfrak {u}}(\mathcal {H})}\) , we may
w.l.o.g. assume that \(x \in C_{{\mathfrak {u}}(\mathcal {H})}\) .
To get more information on \(G/K\) ,
we note that the choice of the complex
structure implies that \(G/K\)  contains
\(\mathop {{\rm Heis}}\nolimits (\mathcal {H})/(K \cap \mathop {{\rm Heis}}\nolimits (\mathcal {H})) \cong \mathcal {H}_-\cong \mathcal {H}^{\prime }\) 
as a complex submanifold.
The translation action of \(\mathop {{\rm Heis}}\nolimits (\mathcal {H})\)  by
\((t,v).z := v + z\)  on \(\mathcal {H}\)  factors through an action of the additive group
\(\mathcal {H}\)  and extends naturally to a holomorphic action of the
complexified group \(\mathcal {H}_\mathcal {H}_+ \times \mathcal {H}_-\) .
As the \(G\) -action on \(G/K\)  induces a transitive
action on the set of \(\mathop {{\rm Heis}}\nolimits (\mathcal {H})\) -orbits on \(G/K\) ,
the action of \(\mathop {{\rm Heis}}\nolimits (\mathcal {H})\)  extends to a holomorphic action of
\(\mathop {{\rm Heis}}\nolimits (\mathcal {H})_ on \) G/K\(. Therefore the action of the one-parameter group\) t (-tx)\( extends to a holomorphic action of \) +\(on \) G/K\( if and only if the same holds for the actionon the quotient space \) Spres(H)/U(H)\(.As we have seen in Example~\ref {ex:liegrp}(h),this space can be identified with the symmetric Hilbert domain\)\( \mathcal {D}_s = \lbrace  z \in B_2(\mathcal {H}) \colon  z^\top = z, zz^* <\!\!< 1 \rbrace , \)\(but here it is endowed with the opposite complex structure.\) The group \(\mathop {\rm U{}}\nolimits (\mathcal {H})\)  acts on \(\mathcal {D}_s\)  by
\(u.z = uzu^\top \) , and for \(x \in C_{{\mathfrak {u}}(\mathcal {H})}\) 
we obtain for our choice of complex structure the relation
\(e^{-ix}.z = e^{ix} ve^{ix^\top }.\) 
Now \(\Vert e^{ix}\Vert  = \Vert e^{ix^\top }\Vert < 1\) 
implies that the action of the one-parameter group
\(t \mapsto \exp (-tx)\)  extends to \(+\) , and this completes the
proof.
Finally, we derive from Theorem REF (b) that
\(I_{\pi _s} = \overline{\mathop {{\rm conv}}\nolimits }(\mathcal {O}_\lambda )\) 
holds for
\( \lambda (x) = - i \langle {\tt d}\pi _s(x)\Omega , \Omega \rangle . \) 
On \({\mathfrak {k}}= \widehat{{\mathfrak {u}}}(\mathcal {H})\)  we have \(\lambda (t,x) = t\) , and \(\lambda \)  vanishes
on \({\mathfrak {p}}_ because \) p-\( annihilates \)\(, and\)ds(p+), = , ds(p-)= 0\(.This completes the proof.\hfill \Box \)
With similar and even easier arguments as in the proof of the
preceding theorem, we also obtain:
Theorem 9.15  
The momentum set \(I_{\pi _s^+}\)  of the even metaplectic representation
\((\pi _s^+, S^{\rm even}(\mathcal {H}))\)  of \(\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\) 
is the closed convex hull of the coadjoint
orbit of \(\Phi _{\pi _s^+}([\Omega ])\) .

The spin representation
In this section we take a closer look
at the spin representation
\((\pi _a, \Lambda (\mathcal {H}))\)  of the
central extension \(\widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})\)  of \(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})\) 
on the fermionic Fock space \(\Lambda (\mathcal {H})\) .
This representation arises from self-intertwining
operators of the Fock representation of the \(C^*\) -algebra
\(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\) . Here we also show that \(\pi _a\)  is semibounded and
determine the corresponding cone \(W_{\pi _a}\) .
For the irreducible representation of the
identity component on the even part
\(\Lambda ^{\rm even}(\mathcal {H})\) , we show that the momentum set
is the weak-\(*\) -closed convex hull of a single coadjoint orbit.
Semiboundedness of the spin representation
Let \(\mathcal {H}\)  be a complex Hilbert space and
write \(\lbrace a,b\rbrace  := ab + ba\)  for the anticommutator of two elements
of an associative algebra. The
CAR-algebra of \(\mathcal {H}\)  is a \(C^*\) -algebra
\(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\) , together with a continuous antilinear map
\(a \colon  \mathcal {H}\rightarrow \mathop {{\rm CAR}}\nolimits (\mathcal {H})\)  satisfying the
canonical anticommutation relations
\(\lbrace a(f), a(g)^*\rbrace  = \langle g,f \rangle \mathbf {1}\quad \hbox{ and } \quad \lbrace a(f),a(g)\rbrace  = 0\quad \mbox{ for } \quad f, g \in \mathcal {H}\) 
and which is generated by the image of \(a\) . This determines
\(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\)  up to natural isomorphism ([32]}).
We also write \(a^*(f) := a(f)^*\) , which defines a complex
linear map \(a^* \colon  \mathcal {H}\rightarrow \mathop {{\rm CAR}}\nolimits (\mathcal {H})\) .
The orthogonal group \(\mathop {\rm O{}}\nolimits (\mathcal {H})\)  of the underlying real
Hilbert space
\((\mathcal {H}_{\mathbb {R}},\beta ), \quad \beta (v,w) = \mathop {{\rm Re}}\nolimits \langle v,w\rangle ,\) 
acts by automorphisms on this \(C^*\) -algebra as follows. Writing
a real linear isometry as
\(g = g_1 + g_2\) , where \(g_1\)  is linear and \(g_2\)  is antilinear, the
relations \(gg^* = g^*g = \mathbf {1}\)  turn into
\( g_1 g_1^* + g_2 g_2^* = \mathbf {1} = g_1^*g_1 + g_2^* g_2\quad \mbox{ and } \quad g_1 g_2^* + g_2 g_1^* = 0 = g_2^* g_1 + g_1^*g_2. \) 
These relations imply that
\( a_g \colon  \mathcal {H}\rightarrow \mathop {{\rm CAR}}\nolimits (\mathcal {H}), \quad f \mapsto a(g_1 f) + a^*(g_2 f) \) 
satisfies the same anticommutation relations, so that the
universal property of \(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\)  implies the existence of a unique
automorphism \(\alpha _g\)  with
\( \alpha _g(a(f)) = a_g(f) \quad \mbox{ for } \quad f \in \mathcal {H}.\) 
These automorphisms of the \(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\)  are called
Bogoliubov automorphisms. They define an action of
\(\mathop {\rm O{}}\nolimits (\mathcal {H})\)  on \(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\) . In particular, the unitary group
\(\mathop {\rm U{}}\nolimits (\mathcal {H}) \subseteq \mathop {\rm O{}}\nolimits (\mathcal {H})\)  acts on \(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\)  by
\(\alpha _g(a(f)) = a(gf)\)  for \(f \in \mathcal {H}\) .
Remark 10.1  
The \(C^*\) -algebra \(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\)  has a natural irreducible
representation
\((\pi _0, \Lambda (\mathcal {H}))\) 
on the antisymmetric Fock space \(\Lambda (\mathcal {H})\)  ([32]}).
The image \(a_0(f) := \pi _0(a(f))\) 
acts by \(a_0(f)\Omega =0\)  and
\( a_0(f)(f_1 \wedge \cdots \wedge f_n)= \sum _{j = 1}^n (-1)^{j-1} \langle f_j, f \rangle f_1 \wedge \cdots \wedge f_{j-1} \wedge f_{j+1} \wedge \cdots \wedge f_n.\) 
Accordingly, we have
\( a_0^*(f)\Omega = f \quad \mbox{ and } \quad a_0^*(f)(f_1 \wedge \cdots \wedge f_n)= f \wedge f_1 \wedge \cdots \wedge f_n. \) 
Let \(Iv = iv\)  denote the complex structure on \(\mathcal {H}\) . The action of the
restricted orthogonal group
\(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})\colon = \lbrace  g \in \mathop {\rm O{}}\nolimits (\mathcal {H}) \colon  \Vert [g,I]\Vert _2 < \infty \rbrace  \) 
on \(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\)  preserves the equivalence class of the representation
\(\pi _0\) ,
so that there is a projective unitary representation
\(\pi _a \colon  \mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H}) \rightarrow \mathop {\rm U{}}\nolimits (\Lambda (\mathcal {H}))\) , called the
spin representation, satisfying
\(\pi _a(g) \pi _0(A)\pi _a(g)^* = \pi _0(\alpha _g A) \quad \mbox{ for }\quad g \in \mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H}), A \in \mathop {{\rm CAR}}\nolimits (\mathcal {H})\) 
(cf. [37]}, [80]}).

In analogy with Theorem REF  we here obtain:
Theorem 10.2 
\(\widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})\)  is a Lie group and
the spin representation is smooth.
A Lie algebra cocycle \(\eta \)  defining \(\widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})\) 
as an extension of \({\mathfrak {o}}_{\rm res}(\mathcal {H})\)  by \({\mathbb {R}}\)  is given by
\( \eta (x,y) = -\frac{1}{2i} \mathop {{\rm tr}}\nolimits ([x_2, y_2]). \) 


Proof.   
With completely analogous arguments as in the proof of
Theorem REF  we find
with Appendix  that any
\(g \in \mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})\)  for which \(g_1\)  is invertible
has a unique lift \(\pi _a(g) \in \mathop {\rm U{}}\nolimits (\Lambda (\mathcal {H}))\)  with
\( \pi _a(g) \Omega = c(g) e^{-\widehat{T}(g)} \quad \mbox{ for } \quad T(g) := g_2 g_1^{-1} \in \mathop {{\rm Aherm}}\nolimits (\mathcal {H})_a,c(g) = \Vert e^{-\widehat{T}(g)}\Vert ^{-1} \) 
(cf. Lemma REF ).
This implies that \(\widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})\)  is a Lie group,
its representation \(\pi _a\)  on \(\Lambda (\mathcal {H})\)  is smooth, and
that \(\Omega \)  is a smooth vector.
Since \(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})\)  acts smoothly on \(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\) ,
the space \(\Lambda (\mathcal {H})^\infty \)  of smooth vectors is invariant under
\(\pi _0(\mathop {{\rm CAR}}\nolimits (\mathcal {H}))\) , so that the irreducibility of the
representation of \(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\)  on \(\Lambda (\mathcal {H})\) 
implies the smoothness of \(\pi _a\) .
For the Lie algebra cocycle defining \(\widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})\) ,
we find as in the proof of Theorem REF 
with Lemma REF (iii)
\(\eta (x,y)= 2 \mathop {{\rm Im}}\nolimits \langle \widehat{x}_2, \widehat{y}_2 \rangle = -\mathop {{\rm Im}}\nolimits \mathop {{\rm tr}}\nolimits (x_2y_2) = -\frac{1}{2i} \mathop {{\rm tr}}\nolimits ([x_2,y_2]).\) 

\(\Box \)
Remark 10.3   As in Subsection REF ,
we obtain a Banach–Lie group
\(\mathop {\rm O{}}\nolimits _{1,2}(\mathcal {H})&:= \Big \lbrace  g = g_1 + g_2 \in \mathop {\rm O{}}\nolimits (\mathcal {H})\colon \Vert g_2 \Vert _2 < \infty , \Vert \mathbf {1} - g_1\Vert _1 < \infty \rbrace \) 

with
\( \mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H}) \cong (\mathop {\rm O{}}\nolimits _{1,2}(\mathcal {H}) \rtimes \mathop {\rm U{}}\nolimits (\mathcal {H}))/N,\quad N \cong \mathop {\rm U{}}\nolimits (\mathcal {H}) \cap \mathop {\rm O{}}\nolimits _{1,2}(\mathcal {H}) = \mathop {\rm U{}}\nolimits _1(\mathcal {H}) \) 
(cf. [8]}). From [8]}
we recall that \(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})\)  has two connected components
and that its fundamental group is trivial.
On the other hand
\( \pi _0(\mathop {\rm O{}}\nolimits _{1,2}(\mathcal {H})) \cong {\mathbb {Z}}/2{\mathbb {Z}}\cong \pi _1(\mathop {\rm O{}}\nolimits _{1,2}(\mathcal {H})) \) 
([83]}, [8]}),
so that there exists a simply connected 2-fold covering group
\(q \colon  {\rm Spin}_{1,2}(\mathcal {H})\rightarrow \mathop {\rm O{}}\nolimits _{1,2}(\mathcal {H})_0\) 
(cf. [85]} for a discussion of the smaller group
\({\rm Spin}_1(\mathcal {H})\)  covering \(\mathop {{\rm SO}}\nolimits _1(\mathcal {H}) = \mathop {\rm O{}}\nolimits _1(\mathcal {H})_0\) ).
As the
inclusion \(\mathop {\rm U{}}\nolimits _{1}(\mathcal {H}) \rightarrow \mathop {\rm O{}}\nolimits _{1,2}(\mathcal {H})\)  induces a surjective homomorphism
\( \pi _1(\mathop {\rm U{}}\nolimits _1(\mathcal {H})) \cong {\mathbb {Z}}\rightarrow \pi _1(\mathop {\rm O{}}\nolimits _{1,2}(\mathcal {H})) \cong {\mathbb {Z}}/2, \) 
\(\widehat{\mathop {\rm U{}}\nolimits }_1(\mathcal {H}) := q^{-1}(\mathop {\rm U{}}\nolimits _1(\mathcal {H}))\)  is the unique
2-fold connected covering of
\(\mathop {\rm U{}}\nolimits _1(\mathcal {H})\)  from Subsection REF .
We further have an embedding
\(\sigma \colon  {\mathfrak {o}}_{1,2}(\mathcal {H}) \rightarrow \widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H}), \quad \sigma (x) := \Big (-\frac{1}{2i} \mathop {{\rm tr}}\nolimits (x_1), x\Big )\) 
of Banach–Lie algebras.
On the subgroup \(\widehat{\mathop {\rm U{}}\nolimits }_1(\mathcal {H}) \subseteq {\rm Spin}_{1,2}(\mathcal {H})\) ,
\(\sigma \)  integrates to a group homomorphism
\(\sigma _G(g) := (g, \sqrt{\det }(g)^{-1}\pi _s(g))\) 
(cf. Subsection REF )
and \(\sigma \)  integrates to a morphism of Banach–Lie groups
\(\sigma _G \colon  {\rm Spin}_{1,2}(\mathcal {H}) \rightarrow \widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H}).\) 
Combining this map with the canonical inclusion
\(\mathop {\rm U{}}\nolimits (\mathcal {H}) \hookrightarrow \widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})\) , the equivariance of
\(\sigma _G\)  under conjugation with unitary operators implies the
existence of a homomorphism
\( \mu \colon  {\rm Spin}_{1,2}(\mathcal {H}) \rtimes \mathop {\rm U{}}\nolimits (\mathcal {H}) \rightarrow \widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H}),\quad (g,u) \mapsto \sigma _G(g)u.\) 

The following proposition is proved as Proposition REF ,
using the representation \(\pi _a\)  instead of \(\pi _s\) .
Proposition 10.4  The homomorphism \(\mu \)  factors through an
isomorphism
\( \overline{\mu }\colon  ({\rm Spin}_{1,2}(\mathcal {H}) \rtimes \mathop {\rm U{}}\nolimits (\mathcal {H}))/\ker \mu \rightarrow \widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})_0\) 
of connected Banach–Lie groups with
\(\ker \mu \cong \mathop {{\rm SU}}\nolimits _1(\mathcal {H}).\)

Remark 10.5  
The embedding \({\mathfrak {o}}_{1,2}(\mathcal {H}) \hookrightarrow \widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})\) 
restricts in particular to an embedding \({\mathfrak {o}}_1(\mathcal {H}) \hookrightarrow \widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})\) . One can show that the operators
\({\tt d}\pi _a(x)\)  are bounded for \(x \in {\mathfrak {o}}_1(\mathcal {H})\) 
(cf. [86]}, [87]}), so that we obtain
a morphism of Banach–Lie algebras
\( {\mathfrak {o}}_1(\mathcal {H}) \rightarrow {\mathfrak {u}}(\Lambda (\mathcal {H})), \quad x \mapsto {\tt d}\pi _a(x) - \frac{1}{2} \mathop {{\rm tr}}\nolimits (x_1)\mathbf {1}. \) 

Definition 10.6  To determine the open cone \(W_{\pi _a}\) 
for the spin representation, we have to take a closer
look at natural cones in \({\mathfrak {o}}_{\rm res}(\mathcal {H})\) .
Let \({\mathfrak {u}}_\infty (\mathcal {H})\)  denote the ideal of compact skew-hermitian
operators in \({\mathfrak {u}}(\mathcal {H})\)  and recall the Calkin algebra
\(\mathop {{\rm Cal}}\nolimits (\mathcal {H}) := B(\mathcal {H})/B_\infty (\mathcal {H})\) , where
\(B_\infty (\mathcal {H})\)  denotes the ideal of compact operators
in the \(C^*\) -algebra \(B(\mathcal {H})\) . Then the surjection
\(B(\mathcal {H}) \rightarrow \mathop {{\rm Cal}}\nolimits (B(\mathcal {H}))\)  induces an isomorphism
\( {\mathfrak {u}}(\mathcal {H})/{\mathfrak {u}}_\infty (\mathcal {H}) \cong {\mathfrak {u}}(\mathop {{\rm Cal}}\nolimits (\mathcal {H})). \) 
Next we note that
\(\lbrace  x \in {\mathfrak {o}}_{\rm res}(\mathcal {H}) \colon  x^* - x \in {\mathfrak {u}}_\infty (\mathcal {H})\rbrace \) 
is a closed ideal of \({\mathfrak {o}}_{\rm res}(\mathcal {H})\) 
which defines a quotient morphism
\(q \colon  {\mathfrak {o}}_{\rm res}(\mathcal {H}) \rightarrow {\mathfrak {u}}(\mathop {{\rm Cal}}\nolimits (\mathcal {H})).\) 
We thus obtain an open invariant cone in \({\mathfrak {o}}_{\rm res}(\mathcal {H})\) 
by
\( W_{{\mathfrak {o}}_{\rm res}(\mathcal {H})} := q^{-1}(C_{{\mathfrak {u}}(\mathop {{\rm Cal}}\nolimits (\mathcal {H}))})= \lbrace  x \in {\mathfrak {o}}_{\rm res}(\mathcal {H}) \colon  -i q(x) <\!\!< 0 \rbrace  \) 
(cf. Example REF ). We write
\(W_{\widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})}\)  for its inverse image in the
central extension \(\widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})\) .

Theorem 10.7  
The spin representation \((\pi _a, \Lambda (\mathcal {H}))\)  of
the connected Lie group \(\widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})\)  is semibounded
with \(W_{\pi _a} = W_{\widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})}\) .

Proof.   
First we note that under
\(\widehat{\mathop {\rm U{}}\nolimits }(\mathcal {H})\cong \mathop {\rm U{}}\nolimits (\mathcal {H})\)  the spin representation decomposes
into the irreducible subspaces \(\Lambda ^n(\mathcal {H})\)  with
\( {\pi _a}(z,g)(v_1 \wedge \cdots \wedge v_n)= z gv_1 \wedge \cdots \wedge g v_n \quad \mbox{ for } \quad v_1,\ldots , v_n \in \mathcal {H}. \) 
If \(i x \le 0\) , then the corresponding operator
on \(\Lambda ^n(\mathcal {H}) \subseteq \mathcal {H}^{\otimes n}\)  is also
\(\le 0\) , and this implies that
\( s_{\pi _a}(t,x) = -t + s_{\pi _a}(x) \le - t < \infty \quad \mbox{ for } \quad x \in C_{{\mathfrak {u}}(\mathcal {H})}. \) 
Writing \({\mathfrak {z}}\cong {\mathbb {R}}\)  for the center of \(\widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})\) ,
we thus find that
\({\mathfrak {z}}\times C_{{\mathfrak {u}}(\mathcal {H})} \subseteq B(I_{\pi _a}).\)
Next we consider the decomposition of \({\mathfrak {g}}= \widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})= {\mathfrak {k}}\oplus {\mathfrak {p}}_2\) , where \({\mathfrak {k}}= {\mathfrak {z}}\times {\mathfrak {u}}(\mathcal {H})\)  and
\({\mathfrak {p}}_2 = \lbrace  x \in {\mathfrak {o}}_{\rm res}(\mathcal {H}) \colon  Ix = - xI\rbrace .\) 
For the map
\( F \colon  {\mathfrak {k}}\times {\mathfrak {p}}_2 \rightarrow {\mathfrak {g}}, \quad (x,y) \mapsto e^{\mathop {{\rm ad}}\nolimits y} x \quad \mbox{ we then have }\quad {\tt d}F(x,0)(a,b) = a + [b,x], \) 
which is invertible if and only if \(\mathop {{\rm ad}}\nolimits x \colon  {\mathfrak {p}}_2 \rightarrow {\mathfrak {p}}_2\)  is
invertible. This is in particular the case for
\(x = I\) , the complex structure of \(\mathcal {H}\) . Therefore the image of
\(({\mathfrak {z}}\times C_{{\mathfrak {u}}(\mathcal {H})}) \times {\mathfrak {p}}_2\)  has interior points,
and since it is contained in the invariant subset
\(B(I_{\pi _a})\) , the cone
\(W_{\pi _a} = B(I_{\pi _a})^0\)  is non-empty.
With Proposition REF  we now see that the
representation \((\pi _a, \Lambda (\mathcal {H}))\)  is semibounded.
To determine the cone \(W_{\pi _a}\) , we recall from Remark REF 
that for each \(x \in {\mathfrak {o}}_1(\mathcal {H})\) , the corresponding
operator \({\tt d}\pi _a(x)\)  on \(\Lambda (\mathcal {H})\)  is bounded. This means that
\({\mathfrak {o}}_1(\mathcal {H}) \subseteq B(I_{\pi _a}) \subseteq \overline{W_{\pi _a}}\) , and
since \(H(W_{\pi _a})\)  is a closed ideal (Lemma REF (v)),
it follows that
\( {\mathfrak {n}}:= {\mathfrak {z}}\oplus {\mathfrak {u}}_\infty (\mathcal {H}) \oplus {\mathfrak {p}}_2 \subseteq H(W_{\pi _a}). \) 
In view of \(\widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H}) = {\mathfrak {u}}(\mathcal {H}) + {\mathfrak {n}}\) , this proves that
\(W_{\widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})} = C_{{\mathfrak {u}}(\mathcal {H})} + {\mathfrak {n}}\subseteq W_{\pi _a}= {\mathfrak {n}}+ (W_{\pi _a} \cap {\mathfrak {u}}(\mathcal {H})). \) 
It therefore remains to show that
\( W_{\pi _a} \cap {\mathfrak {u}}(\mathcal {H}) \subseteq C_{{\mathfrak {u}}(\mathcal {H})} + {\mathfrak {u}}_\infty (\mathcal {H}). \) 
To verify this assertion, let \(A = A^*\)  be a hermitian
operator on \(\mathcal {H}\)  and \({\tt d}\pi _a(A)\)  denote the corresponding operator
on \(\Lambda ^n(\mathcal {H})\) . Suppose that there exists an \(\varepsilon > 0\) 
such that the range of the spectral projection
\(P([\varepsilon ,\infty [)\)  is infinite dimensional. Then there exists
for each \(n\)  an orthonormal subset \(v_1, \ldots , v_n\)  in this space.
We then have
\(&\ \ \ \ \langle {\tt d}\pi _a(A)(v_1 \wedge \cdots \wedge v_n),v_1 \wedge \cdots \wedge v_n \rangle \\&= \sum _{j = 1}^n \langle v_1 \wedge \cdots \wedge Av_j \wedge \cdots \wedge v_n,v_1 \wedge \cdots \wedge v_n \rangle = \sum _{j = 1}^n \langle Av_j, v_j \rangle \ge n \varepsilon ,\) 

and this implies that \({\tt d}\pi _a(A)\)  is not bounded from above.
If, conversely, for every \(\varepsilon > 0\) 
the spectral projection \(P([\varepsilon ,\infty [)\)  has finite dimensional
range, then \(A = A_- + A_+\) , where
\(A_- \le 0\)  and \(A_+\)  is compact, i.e.,
\(iA \in \overline{C_{{\mathfrak {u}}(\mathcal {H})}} + {\mathfrak {u}}_\infty (\mathcal {H})\) .
This implies that the open cone \(W_{\pi _a} \cap {\mathfrak {u}}(\mathcal {H})\) 
is contained in \(C_{{\mathfrak {u}}(\mathcal {H})} + {\mathfrak {u}}_\infty (\mathcal {H})\) .
\(\Box \)
Remark 10.8  
(a) With a similar argument as in Remark REF  ones
argues that under the action of the identity component
\(\widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})_0\)  on the space \(\Lambda (\mathcal {H})\) 
decomposes into two irreducible subrepresentations
\(\Lambda ^{\rm even}(\mathcal {H})\)  and \(\Lambda ^{\rm odd}(\mathcal {H})\) .
However, the action of the full group \(\widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})\) 
is irreducible because the elements \(g \in \widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})\) 
not contained in the identity component exchange
the two subspaces
\(\Lambda ^{\rm even}(\mathcal {H})\)  and
\(\Lambda ^{\rm odd}(\mathcal {H})\)  ([83]}).
(b) There is also an analog of the Banach–Lie algebra
\(\mathfrak {hsp}_{\rm res}(\mathcal {H})\)  acting irreducibly on \(\Lambda (\mathcal {H})\) 
in the fermionic case. Here it is an infinite dimensional
analog of the odd orthogonal Lie algebra \(\mathop {{\mathfrak {so} }}\nolimits _{2n+1}({\mathbb {R}})\) .
To see this Lie algebra, we define linear and antilinear
rank-one operators on \(\mathcal {H}\)  by
\( P_{v,w}(x) := \langle x,w \rangle v \quad \mbox{ and } \quad \overline{P}_{v,w}(x) := \langle w,x\rangle v \) 
and observe that \(P_{v,w}^* = P_{w,v}\)  and
\(\overline{P}_{v,w}^* = \overline{P}_{w,v}\) .
Therefore \(Q_{v,w} := P_{v,w} - P_{w,v} \in {\mathfrak {u}}_1(\mathcal {H})\) ,
and a direct calculation
yields
\( {\tt d}\pi _a(Q_{v,w}) = a_0^*(v)a_0(w) - a_0^*(w)a_0(v). \) 
As \(\mathop {{\rm tr}}\nolimits (Q_{v,w}) = 2i \mathop {{\rm Im}}\nolimits \langle v,w\rangle \) , we obtain with the map
\(\sigma \)  from (REF ) (Remark REF )
\( {\tt d}\pi _a(\sigma (Q_{v,w})) = {\tt d}\pi _a(-\mathop {{\rm Im}}\nolimits \langle v,w \rangle , Q_{v,w}). \) 
For the antilinear operators \(\overline{Q}_{v,w} :=\overline{P}_{v,w} - \overline{P}_{w,v}\)  we have
\(\overline{Q}_{v,w} \in {\mathfrak {o}}_1(\mathcal {H}) \subseteq {\mathfrak {o}}_{1,2}(\mathcal {H})\)  and
\( {\tt d}\pi _a(\overline{Q}_{v,w})= a_0^*(v)a_0^*(w) - a_0(w) a_0(v) = a_0^*(v)a_0^*(w) + a_0(v) a_0(w). \) 
Next we observe that the operators \(\rho (v) := \frac{1}{\sqrt{2}}(a_0(v) -a_0^*(v))\)  satisfy
\(&\ \ \ \ [\rho (v), \rho (w)]= {\textstyle {\frac{1}{2}}}[a_0(v) - a_0^*(v), a_0(w) - a_0^*(w)] \\&= (a_0(v)a_0(w) + a_0^*(v)a_0^*(w))+ (a_0^*(w) a_0(v) - a_0^*(v)a_0(w)) + i\mathop {{\rm Im}}\nolimits \langle v, w \rangle \\&= {\tt d}\pi _a(\mathop {{\rm Im}}\nolimits \langle v, w \rangle , \overline{Q}_{v,w}-Q_{v,w})={\tt d}\pi _a(\sigma (\overline{Q}_{v,w}- Q_{v,w})).\) 

This calculation implies that we obtain on the direct sum
\(\mathcal {H}\oplus {\mathfrak {o}}_{1,2}(\mathcal {H})\)  a Banach–Lie algebra structure
with the bracket
\( [(v,X), (v^{\prime },X^{\prime })]:= (Xv^{\prime } - X^{\prime }v, [X,X^{\prime }] + \overline{Q}_{v,w}-Q_{v,w}), \) 
and
\( (v,X) \mapsto \rho (v) + {\tt d}\pi _a(\sigma (X)) \) 
defines a representation by unbounded operators on \(\Lambda (\mathcal {H})\) ,
where the subalgebra \(\mathcal {H}\oplus {\mathfrak {o}}_1(\mathcal {H})\)  is represented by bounded
operators. A closer inspection shows that
\(\mathcal {H}\oplus {\mathfrak {o}}_1(\mathcal {H}) \cong {\mathfrak {o}}_1(\mathcal {H}\oplus {\mathbb {R}})\)  is an
infinite dimensional version of \({\mathfrak {o}}_{2n+1}({\mathbb {R}})\) 
(cf. [87]}).

Definition 10.9  Let \({\mathcal {I}}_\beta \subseteq \mathop {{\rm GL}}\nolimits (\mathcal {H}_{\mathbb {R}})\)  be the set of
orthogonal real-linear complex structures on the real
Hilbert space \(\mathcal {H}_{\mathbb {R}}\) . This set parameterizes the
complex Hilbert space structures on \(\mathcal {H}_{\mathbb {R}}\)  compatible
with the given real Hilbert space structure.

Lemma 10.10  The following assertions hold:
(i)
\({\mathcal {I}}_\beta = \mathop {\rm O{}}\nolimits (\mathcal {H}) \cap {\mathfrak {o}}(\mathcal {H})= \lbrace  g \in \mathop {{\rm GL}}\nolimits (\mathcal {H}_{\mathbb {R}}) \colon  g^\top = g^{-1} = -g \rbrace \)  is a submanifold
of \(\mathop {\rm O{}}\nolimits (\mathcal {H})\) .
(ii)
The conjugation action of \(\mathop {\rm O{}}\nolimits (\mathcal {H})\)  on
\({\mathcal {I}}_\beta \)  leads to a diffeomorphism \({\mathcal {I}}_\beta \cong \mathop {\rm O{}}\nolimits (\mathcal {H})/\mathop {\rm U{}}\nolimits (\mathcal {H})\) .
(iii)
\({\mathcal {I}}_\beta ^{\rm res} := \mathop {{\rm Ad}}\nolimits (\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H}))I= \lbrace  J \in {\mathcal {I}}_\beta \colon  \Vert I - J \Vert _2 < \infty \rbrace \cong \mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})/\mathop {\rm U{}}\nolimits (\mathcal {H})\) .


Proof.   
(i) It only remains to show that \({\mathcal {I}}_\beta \)  is a
submanifold of \(\mathop {\rm O{}}\nolimits (\mathcal {H})\) . For \(J \in {\mathcal {I}}_\beta \)  we parameterize
a neighborhood of \(J\)  by the map
\({\mathfrak {o}}(\mathcal {H}) \rightarrow \mathop {\rm O{}}\nolimits (\mathcal {H}),x \mapsto J e^x\) 
which is a diffeomorphism on some open 0-neighborhood \(U\subseteq {\mathfrak {o}}(\mathcal {H})\) , which we may assume to be invariant under
\(\mathop {{\rm Ad}}\nolimits (J)\) . That \(J e^x\)  is a complex structure is equivalent to
\( -Je^x = (Je^x)^{-1} = e^{-x} J^{-1} = - J e^{-J^{-1}xJ}, \) 
which is equivalent to
\(Jx = - xJ\) . We conclude that \({\mathcal {I}}_\beta \)  is a submanifold
of \(\mathop {\rm O{}}\nolimits (\mathcal {H})\)  whose tangent space can be identified
with the set of \(J\) -antilinear elements in \({\mathfrak {o}}(\mathcal {H})\) .
(ii) Let \(I \in {\mathcal {I}}_\beta \)  be the canonical complex structure
given by \(Iv = iv\)  and \(J \in {\mathcal {I}}_\beta \) .
Then \(\mathcal {H}= (\mathcal {H}_{\mathbb {R}}, I)\)  and \((\mathcal {H}_{\mathbb {R}}, J)\)  are two complex Hilbert
spaces whose underlying real Hilbert spaces are isomorphic.
This implies that they have the same complex Hilbert dimension,
i.e., there exists a unitary isomorphism
\(g \colon  (\mathcal {H}_{\mathbb {R}}, I) \rightarrow (\mathcal {H}_{\mathbb {R}}, J)\) , i.e., \(g \in \mathop {\rm O{}}\nolimits (\mathcal {H})\)  with
\(gI = Jg\) . Therefore \(\mathop {\rm O{}}\nolimits (\mathcal {H})\)  acts transitively on
\({\mathcal {I}}_\beta \)  and the stabilizer of \(I\)  is the subgroup
\(\mathop {\rm U{}}\nolimits (\mathcal {H})\)  (cf. [90]}).
Since its Lie algebra \({\mathfrak {u}}(\mathcal {H})\)  is complemented by
the closed subspace
\({\mathfrak {p}}:= \lbrace  x \in {\mathfrak {o}}(\mathcal {H}) \colon  Ix = - xI\rbrace ,\) 
it follows that \(\mathop {\rm O{}}\nolimits (\mathcal {H})/\mathop {\rm U{}}\nolimits (\mathcal {H})\)  is a Banach homogeneous space
diffeomorphic to \({\mathcal {I}}_\beta \) .
(iii) For \(g = g_1 + g_2 \in \mathop {\rm O{}}\nolimits (\mathcal {H})\)  the operator
\([I,g] = [I, g_2] = 2 I g_2\)  is Hilbert–Schmidt if and only if
\(g_2\)  is, i.e., \(g \in \mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})\) .
This in turn is equivalent to
\(gIg^{-1} - I = [g,I]g^{-1}\)  being Hilbert–Schmidt, so that
\( \mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H}) = \lbrace  g \in \mathop {\rm O{}}\nolimits (\mathcal {H}) \colon \Vert gIg^{-1} - I \Vert _2 < \infty \rbrace  \) 
and therefore (ii) implies (iii).
\(\Box \)
Remark 10.11  A priori, the \(C^*\) -algebra
\(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\)  depends on the complex structure on \(\mathcal {H}\) , but
it can also be expressed as the \(C^*\) -algebra generated by
the hermitian elements \(b(f) := \frac{1}{\sqrt{2}}(a(f) + a^*(f))\)  satisfying
\( \lbrace b(f), b(g)\rbrace = {\textstyle {\frac{1}{2}}}\lbrace a(f), a^*(g)\rbrace  + {\textstyle {\frac{1}{2}}}\lbrace a^*(f), a(g)\rbrace = {\textstyle {\frac{1}{2}}}(\langle g, f \rangle + \langle f, g\rangle ) \mathbf {1} = \beta (f,g) \mathbf {1}. \) 
In this sense it is the \(C^*\) -envelope of the Clifford algebra
of \((\mathcal {H}_{\mathbb {R}},\beta )\) .
From this point of view it is even more transparent why the group
\(\mathop {\rm O{}}\nolimits (\mathcal {H})\)  acts by automorphisms.
Now we can think of the Fock representation \(\pi _0\)  as
depending on the complex structure \(I\)  on \(\mathcal {H}\) , and any other
representation \(\pi _0 \circ \alpha _g\)  is the Fock representation
corresponding to the complex structure \(gIg^{-1} \in {\mathcal {I}}_\beta \) .
We thus obtain a map of \({\mathcal {I}}_\beta \)  into the set \(\mathop {{\rm Ext}}\nolimits (S(\mathop {{\rm CAR}}\nolimits (\mathcal {H})))\) 
of pure states of \(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\) , mapping
\(J := gIg^{-1}\)  to \(\omega _J(A) := \langle \pi _0(\alpha _g^{-1}(A))\Omega ,\Omega \rangle \) . These states are called the Fock states of
\(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\) , and in [90]} they are essentially
characterized as the pure quasi-free states (cf. Subsection REF  below).

Theorem 10.12   For the irreducible subrepresentation
\((\pi _a^+, \Lambda ^{\rm even}(\mathcal {H}))\)  of \(\widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})\) 
the momentum set is the closed convex hull of the coadjoint orbit of
\(\lambda := \Phi _{\pi _a}([\Omega ])\) . This functional is given
by
\(\lambda (t,x) = t\quad \mbox{ on } \quad \widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H}) = {\mathbb {R}}\oplus {\mathfrak {o}}_{\rm res}(\mathcal {H}). \) 


Proof.   
We want to apply Theorem REF .
For the subgroup \(K := \widehat{\mathop {\rm U{}}\nolimits }(\mathcal {H})\subseteq G := \widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})_0\) 
with Lie algebra
\({\mathfrak {k}}= {\mathfrak {z}}\oplus {\mathfrak {u}}(\mathcal {H})\) , we have
\( {\mathfrak {g}}_{\mathfrak {p}}_+ \oplus {\mathfrak {k}}_{\mathfrak {p}}_-,\quad \mbox{ where } \quad {\mathfrak {p}}_\pm = \ker (\mathop {{\rm ad}}\nolimits I \mp 2i\mathbf {1}) \) 
and \(I \in {\mathfrak {u}}(\mathcal {H})\)  is the multiplication with \(i\)  on \(\mathcal {H}\) .
Since the Lie algebra \({\mathfrak {k}}\)  of \(K\)  is complemented by the closed
subspace \({\mathfrak {p}}_2\)  of antilinear elements in \({\mathfrak {g}}\) , the coset space
\(G/K\)  carries the structure of a Banach homogeneous space
(cf. Lemma REF ).
Moreover, the closed \(K\) -invariant subalgebra
\({\mathfrak {k}}_{\mathfrak {p}}_+\)  determines on \(G/K\)  the structure of a
complex manifold for which the tangent space in the base point
can be identified with \({\mathfrak {g}}_({\mathfrak {k}}_ {\mathfrak {p}}_+) \cong {\mathfrak {p}}_-\) 
([76]}).
In view of (REF ), we obtain with
[8]} that the
group \(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})\)  acts transitively on the homogeneous
space \(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H}_\beta _/P\) ,
where \(P\)  is the stabilizer of the subspace \(\mathcal {H}\oplus \lbrace 0\rbrace \subseteq \mathcal {H}_ (cf.\ Example~\ref {ex:liegrp}(d),(i)).From \) P Ores(H) = U(H)\( it now follows thatthe action of the Banach--Lie group \) G\( on \) G/K\( actually extends to aholomorphic action of a complex group.\) Let \(\Omega \in \Lambda ^0(\mathcal {H})\)  be the vacuum vector.
Then \(\Omega \)  is a smooth vector
which is an eigenvector for \(K\) . On \(\Lambda (\mathcal {H})\) , the operator
\(-i{\tt d}\pi (I)\)  is diagonal, and \(\Lambda ^n(\mathcal {H})\)  is the eigenspace
corresponding to the eigenvalue \(n\) . As \(-i\mathop {{\rm ad}}\nolimits I\)  acts on \({\mathfrak {p}}_-\)  with the
eigenvalue \(-2\) , it follows that
\({\tt d}\pi _a({\mathfrak {p}}_-)\Omega = \lbrace 0\rbrace ,\) 
so that Theorem REF  provides a holomorphic equivariant map
\( \eta \colon  G/K\rightarrow {\mathbb {P}}(\Lambda (\mathcal {H})^{\prime }), \quad g \mapsto \langle \cdot , \pi _a(g)^{-1} \Omega \rangle \) 
and a realization of \((\pi _a, \Lambda ^{\rm even}(\mathcal {H}))\) 
by holomorphic sections of a complex line bundle over \(G/K\) .
Since the representation
\((\pi _a, \Lambda ^{\rm even}(\mathcal {H}))\)  of \(G\) 
is irreducible, \(\Omega \)  is a cyclic vector.
As we have argued above, the action of every one-parameter subgroup of
\(G\)  extends to a holomorphic action of \( on \) G/K\(,so that Theorem~\ref {thm:5.7}(c) implies that\) I\( is the closed convex hull of the coadjoint orbitof \)\(. That \)\( has the desired form followsas in the proof of Theorem~\ref {thm:9.14}.\hfill \Box \)
Quasi-free representations
Let \(P = P^* = P^2\)  be an
orthogonal projection on \(\mathcal {H}\) ,
\(\mathcal {H}_- := \mathop {{\rm im}}\nolimits P\) , \(\mathcal {H}_+ := \ker P\) , and
\(\Gamma \colon  \mathcal {H}\rightarrow \mathcal {H}\)  be an isometric antilinear involution
commuting with \(P\) . Then
\(\tau _P := (\mathbf {1} - P) + \Gamma P \in \mathop {\rm O{}}\nolimits (\mathcal {H})\) .
We write \(a_0 \colon  \mathcal {H}\rightarrow B(\Lambda (\mathcal {H}))\)  for the map corresponding to
the Fock representation of \(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\) 
(Remark REF ).
Twisting with the Bogoliubov automorphism defined by \(\tau _P\) , we
obtain an irreducible
representation \((\pi _P, \Lambda (\mathcal {H}))\)  of \(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\)  by
\( a_P(f) := a_0((\mathbf {1}-P)f) + a_0^*(\Gamma Pf)= \pi _0(a_{\tau _P}(f)) \quad \mbox{ for }\quad f \in \mathcal {H}, \) 
i.e., \(\pi _P = \pi _0 \circ \alpha _{\tau _P}\) .
These representations are called quasi-free.
For \(P =0\)  we recover the Fock representation defined by \(a_0\) .
Two quasi-free representations \(a_P\)  and \(a_Q\)  are equivalent if and only if
\(\Vert P - Q\Vert _2 < \infty \)  ([94]}).
Remark 10.13  The physical interpretation of \(P\)  is that its
range \(\mathcal {H}_-\)  consists
of the negative energy states and its kernel \(\mathcal {H}_+\)  consists of the
positive energy states.
For \(f \in \mathcal {H}_+\)  we have
\(a_P(f)\Omega = 0\)  and likewise
\(a_P^*(f)\Omega = 0\)  for \(f \in \mathcal {H}_-\) .
This is interpreted in such a way that the
creation operator \(a_P^*(f)\)  cannot create an additional negative
state from \(\Omega \)  because all negative states are already
filled (Pauli's Principle).
Likewise, the annihilation operator \(a_P(f)\) , corresponding
to the positive energy vector \(f\) , cannot extract any positive
energy state from \(\Omega \) .

The restricted unitary group \(\mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)\) 
(cf. Example REF (c)) is a subgroup of \(\mathop {\rm O{}}\nolimits (\mathcal {H})\) ,
and for all projections \(Q = gPg^{-1}\) , \(g \in \mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)\) 
we have \(\Vert P - Q\Vert _2 < \infty \) , so that the equivalence of
\(a_P\)  and \(a_Q\)  leads to a projective unitary representation
of \(\mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H})\)  on \(\Lambda (\mathcal {H})\)  determined by
\( \pi _a^P(g) a_P(f) \pi _a^P(g)^* = a_P(gf), \quad f \in \mathcal {H},g \in \mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H}). \) 
Let \(\widehat{\mathop {\rm U{}}\nolimits }_{\rm res}(\mathcal {H})\)  denote the corresponding central
extension and write \(\pi _a^P\)  for its unitary representation
on \(\Lambda (\mathcal {H})\) . To see that we thus obtain a semibounded
representation of a Lie group, we first note that
we have an embedding
\(\iota \colon  \mathop {\rm U{}}\nolimits (\mathcal {H}) \rightarrow \mathop {\rm O{}}\nolimits (\mathcal {H}), \quad g \mapsto \tau _P g \tau _P \) 
for which \(\mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)\)  is precisely the inverse image
of \(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})\)  and that for \(g \in \mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)\)  we have
\(\pi _a(\tau _P g \tau _P)a_P(f)\pi _a(\tau _P g \tau _P)^*&= \pi _a(\tau _P g \tau _P)\pi _0(\alpha _{\tau _P} a(f))\pi _a(\tau _P g \tau _P)^*\\&= \pi _0(\alpha _{\tau _P g} a(f)) = \pi _P(a_g(f)) = a_P(gf),\) 

so that the projective representation of \(\mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)\)  on
\(\Lambda (\mathcal {H})\)  coincide with \(\pi _a \circ \iota \) 
(cf. [37]}). It follows in particular that
\(\widehat{\mathop {\rm U{}}\nolimits }_{\rm res}(\mathcal {H},P) \cong \iota ^*\widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})\) 
is a Lie group, \(\pi _a^P\)  is a smooth representation,
and since \(\mathop {{\rm im}}\nolimits (\mathop {\bf L{}}\nolimits (\iota ))\)  intersects the cone \(C_{{\mathfrak {u}}(\mathcal {H})}\subseteq W_{\pi _a}\) , the representation \(\pi _a^P\)  is semibounded
(cf. Proposition REF (iv)).
More precisely, each element \(x = (x_+, x_-) \in {\mathfrak {u}}(\mathcal {H}_+) \oplus {\mathfrak {u}}(\mathcal {H}_-)\)  with
\(i x_+ <\!\!< 0\)  and \(ix_- >\!\!> 0\)  lies in \(W_{\pi _a^P}\) .
This holds in particular for \(\tau _p I \tau _p = I(\mathbf {1} - 2P)\) .
In the physics literature, the
corresponding selfadjoint operator
\(Q := -i {\tt d}\pi _a^P(I) = {\tt d}\pi _a(\mathbf {1} - 2P)\) 
is called the charge operator and \(\mathbf {1} - 2P\)  the one-particle charge operator.
Its 1-eigenspace is \(\mathcal {H}_+\)  and its \(-1\) -eigenspace is \(\mathcal {H}_-\) .
Remark 10.14  The situation described above can be viewed as a
“second quantization” procedure that can be used to
turn a self-adjoint operator \(A\)  on the single particle space
\(\mathcal {H}\)  into a non-negative operator \(\widehat{A}\)  on the many particle
space. In fact, let \(P := P(]-\infty ,0[)\)  denote the
spectral projection of \(A\)  corresponding to the open negative axis.
Then \(iA\)  generates a strongly continuous one-parameter group
\(\gamma _A(t) := e^{itA}\) 
of \(\mathop {\rm U{}}\nolimits (\mathcal {H}_+) \times \mathop {\rm U{}}\nolimits (\mathcal {H}_-) \subseteq \mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)\) ,
and \(\pi _a^P(\gamma _A(t)) = e^{it \widehat{A}}\)  is a one-parameter group
of \(\mathop {\rm U{}}\nolimits (\Lambda (\mathcal {H}))\)  whose infinitesimal generator
\(\widehat{A}\)  has non-negative spectrum.

For more details on the complex manifolds
\(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})/\mathop {\rm U{}}\nolimits (\mathcal {H})\)  (the isotropic restricted Graßmannian)
and \(\mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)/(\mathop {\rm U{}}\nolimits (P(\mathcal {H})) \times \mathop {\rm U{}}\nolimits (\mathcal {H}_+))\) 
(the restricted or Sato–Segal–Wilson Graßmannian) we refer to
[83]} and [97]}, where one also finds a detailed
discussion of the corresponding complex line bundles,
the Pfaffian line bundle \({\rm Pf}\)  over \(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})/\mathop {\rm U{}}\nolimits (\mathcal {H})\) 
whose dual permits the even spin representation as a space of
holomorphic sections, and the determinant line bundle \({\rm Det}\) 
over the restricted Graßmannian whose dual provides a
realization of the representations of \(\widehat{\mathop {\rm U{}}\nolimits }_{\rm res}(\mathcal {H},P)\) 
mentioned above. Physical aspects of highest weight representations
of \(\mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H})\)  are discussed in [98]}.

Perspectives
Classification problems:
In the preceding sections we mainly discussed three
prototypical classes of semibounded representations:
highest weight representations of the Virasoro group,
the metaplectic representation on the bosonic Fock space
and the spin representation on the fermionic Fock space.
These representations are of fundamental importance in
mathematical physics and homomorphisms from a Lie group
\(G\)  to \(\mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})\) , \(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})\)  or
\(\mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)\) 
can be used to obtain semibounded representations of a central
extension by pulling back the representations discussed above.
This construction has been a major source of (projective) representations
for loop groups \(G = C^\infty ({\mathbb {S}}^1,K)\)  and \(\mathop {{\rm Diff}}\nolimits ({\mathbb {S}}^1)\) 
(cf. [35]}, [83]}, [101]}, [98]}).
What still remains to be developed is a better global perspective
on semibounded representations, including classification
results on irreducible ones and the existence of direct integral
decompositions.
Problem 11.1  Classify all irreducible semibounded unitary representations
of the groups \(\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\) , \(\widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})\) 
and \(\widehat{\mathop {\rm U{}}\nolimits }_{\rm res}(\mathcal {H},P)\) , where \(\mathcal {H}\)  is a complex Hilbert space and
\(P\)  an orthogonal projection on \(\mathcal {H}\) .
These classification problems are special cases of the more general
problem of the classification of the projective semibounded
unitary representations of the automorphism groups
of hermitian Hilbert symmetric spaces. For the three groups above,
the corresponding spaces of
\({\mathcal {I}}_\omega ^{\rm res}\) , \({\mathcal {I}}_\beta ^{\rm res}\) , resp.,
the restricted Graßmannian \(\mathop {{\rm Gr}}\nolimits _{\rm res}(\mathcal {H},P) :=\lbrace  gPg^{-1} \colon  g \in \mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)\rbrace \) . A crucial
difference between these spaces is that the first one
is equivalent to a symmetric Hilbert domain
(of negative curvature) and the latter two are positively
curved spaces. This difference is also reflected in the difference
between the cones \(W_{\pi _s}\)  and \(W_{\pi _s}\) 
in \(\widehat{\mathfrak {sp}}_{\rm res}(\mathcal {H})\)  and \(\widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})\) 
(Theorems REF  and REF ).
An even larger class of groups arises as automorphism groups
of Hilbert flag manifolds such as the orbit of a finite
flag in \(\mathcal {H}\)  under the group \(\mathop {\rm U{}}\nolimits _2(\mathcal {H})\) . For a systematic
discussion of these manifolds and the topology of the
corresponding real and complex groups we refer to
[8]} and for corresponding representations
to [104]}.

For loop groups of the form \(C^\infty ({\mathbb {S}}^1,K)\) ,
\(K\)  a compact connected Lie group, the irreducible
projective positive energy representations can be identified
as highest weight representations ([83]},
[106]}, and in particular
[35]} for \(K = ).With the convexity theorems in \cite {AP83} and \cite {KP84} it shouldbe possible to show that these representationsare semibounded and one can hope for an analog ofTheorem~\ref {thm:8.22} asserting that every irreduciblesemibounded representation of the corresponding double extensioneither is a highest weight representation or its dual.In \cite {Ne09b} it is shown on the algebraic level that therealso exist many interesting unitary representationsof infinite rank generalizations of twisted and untwistedloop algebras, resp., their double extensions(so-called locally affine Lie algebras). On the group levelthey correspond to (double extensions of) groupsof the form \) C(S1, K)\(, where\) K\( is a Hilbert--Lie group, such as \) U2(H)\(.It seems quite likely that all these representationsare semibounded with momentum sets generated by a singlecoadjoint orbit.\)New sources of semibounded representations:
Although many interesting classes of semibounded representations
are known, a systematic understanding of the geometric sources
of these representations is still lacking.
As we have seen above, on the Lie algebra level the existence
of open invariant cones in a trivial central extension
is necessary (Remark REF ). However, as the example
\(\mathcal {V}({\mathbb {S}}^1)\)  shows, it is not sufficient
(Theorem REF ).
Clearly, the circle \({\mathbb {S}}^1\)  plays a special role in many constructions,
as the rich theory for central extensions of loop groups and the
Virasoro group shows. Beyond \({\mathbb {S}}^1\) , it seems that
Lie algebras of conformal vector fields
(as generalizations of \(\mathcal {V}({\mathbb {S}}^1)\) ) (cf. Example REF , [108]} and [109]}) and Lie algebras of
sections of vector bundles over Lorentzian manifolds
(or more general “causal” spaces, [110]})
are natural candidates to be investigated
with respect to the existence of semibounded representations.
The latter class of Lie algebras is a natural
generalization of loop algebras.
Here an interesting point is that, although the conformal
groups \(\mathop {\rm Conf{}}\nolimits ({\mathbb {S}}^1) = \mathop {{\rm Diff}}\nolimits ({\mathbb {S}}^1)\)  of the circle and the conformal
group \(\mathop {\rm Conf{}}\nolimits ({\mathbb {S}}^{1,1})\)  of the Lorentzian torus
are infinite dimensional ,
for Lorentzian manifolds of dimension \(\ge 3\)  the conformal groups are finite
dimensional (cf. [111]}). In particular, it is contained in the list
of [108]}. This leads to the well-studied class of hermitian
Lie groups (see in particular [109]}, [114]}, [110]}).
Coadjoint orbits: There is a symplectic version of semibounded
representations, namely Hamiltonian actions
\(\sigma \colon  G \times M \rightarrow M\)  with a momentum map
\(\Phi \colon  M \rightarrow {\mathfrak {g}}^{\prime }\)  for which the image of the momentum map
is semi-equicontinuous. If \(G\)  is finite dimensional
and \(\Phi (M)\)  is closed, then Proposition REF 
implies that there exists an \(x \in {\mathfrak {g}}\)  for which the
Hamiltonian function \(H_x(m) := \Phi (m)(x)\)  is proper.
In particular, \(\Phi \)  is a proper map. In this sense
the semi-equicontinuity of the image of \(\Phi \)  is
a weakening of properness, which is a useful property
as far as convexity properties
are concerned ([116]}).
Even though we do not know in general to which extent coadjoint
orbits of infinite dimensional Lie groups are manifolds, the orbit
\(\mathcal {O}_\lambda \)  can always be viewed as the range of a momentum map
\(\Phi \colon  G \rightarrow {\mathfrak {g}}^{\prime }, g \mapsto \mathop {{\rm Ad}}\nolimits ^*(g)\lambda \)  corresponding to the
left action of \(G\)  on itself preserving the left invariant
closed 2-form \(\Omega \)  with \(\Omega _1(x,y) = \lambda ([x,y])\) .
This action is semibounded if and only if \(\mathcal {O}_\lambda \)  is
semi-equicontinuous.
Since the momentum sets of semibounded representations always consists
of semi-equicontinuous orbits, the identification of the set
\({\mathfrak {g}}^{\prime }_{\rm seq}\)  of semi-equicontinuous coadjoint orbits of a given
Lie algebra is already a solid first step towards the understanding
of corresponding semibounded representations.
As we have seen in many situations above, a useful tool to
study convexity properties of coadjoint orbits
are projection maps
\(p_{\mathfrak {t}}\colon  {\mathfrak {g}}\rightarrow {\mathfrak {t}}\) , where \({\mathfrak {t}}\subseteq {\mathfrak {g}}\)  is a
“compactly embedded” subalgebra, i.e.,
the action of \(e^{\mathop {{\rm ad}}\nolimits {\mathfrak {t}}}\)  on \({\mathfrak {g}}\)  factor through
the action of a compact abelian group.
As we have seen in Section , sometimes one
does not want to project to abelian subalgebras and
one has to study projections \(p_{\mathfrak {k}}\colon  {\mathfrak {g}}\rightarrow {\mathfrak {k}}\) ,
where \({\mathfrak {k}}\)  is a subalgebra for which \(e^{\mathop {{\rm ad}}\nolimits {\mathfrak {k}}}\) 
leaves a norm on \({\mathfrak {g}}\)  invariant
This is of particular interest to understand open invariant
cones \(W \subseteq {\mathfrak {g}}\)  because they often have the form
\( W = \mathop {{\rm Ad}}\nolimits (G) (W \cap {\mathfrak {k}}) \quad \mbox{ with } \quad p_{\mathfrak {k}}(W) = W \cap {\mathfrak {k}}\) 
(cf. Theorem REF ).
In this situation on needs convexity theorems of the type
\( p_{\mathfrak {k}}(\mathcal {O}_x) \subseteq \overline{\mathop {{\rm conv}}\nolimits }(\mathop {{\rm Ad}}\nolimits (N_G({\mathfrak {k}}))x) + C, \) 
where \(C\)  is a certain invariant convex cone in \({\mathfrak {k}}\)  and
\(N_G({\mathfrak {k}}) \subseteq G\)  is the normalizer of \({\mathfrak {k}}\)  in \(G\) 
(cf. [1]} for finite dimensional Lie algebras).
If \({\mathfrak {k}}= {\mathfrak {t}}\)  is abelian, then \(N_G({\mathfrak {t}})\)  is an analog
of the Weyl group.
For infinite dimensional Lie algebras, not many convexity theorems
are known. Of relevance for semibounded representations
is the particular case of affine Kac–Moody Lie algebras
([118]}, [119]}), and for Lie algebras of bounded
operators on Hilbert spaces
the infinite dimensional version of Kostant's Theorem by
A. Neumann ([120]}) is crucial. What is still lacking
is a uniform framework for results of this type.

Smooth vectors for representations
Let \(G\)  be a Lie group with Lie algebra \({\mathfrak {g}}\)  and exponential
function \(\exp _G \colon  {\mathfrak {g}}\rightarrow G\) .
Further, let \(V\)  be a locally convex space and
\(\pi \colon  G \rightarrow \mathop {{\rm GL}}\nolimits (V)\)  be a homomorphism
defining a continuous action of \(G\)  on \(V\) . We write
\(\pi ^v(g) := \pi (g)v\)  for the orbit maps and
\(V^\infty := \lbrace  v \in V \colon  \pi ^v \in C^\infty (G,V) \rbrace  \) 
for the space of smooth vectors.
In this appendix we collect some results of
[30]} that are used in the present paper.
Let
\({\tt d}\pi \colon  {\mathfrak {g}}\rightarrow \mathop {{\rm End}}\nolimits (V^\infty ), \quad {\tt d}\pi (x)v := \frac{d}{dt} \hbox{\vrule width0.5ptheight 5mm depth 3mm${{}\atop {{}\atop {\scriptstyle t=0}}}$} \pi (\exp tx)v \) 
denote the derived action of \({\mathfrak {g}}\)  on \(V^\infty \) .
That this is indeed a representation of \({\mathfrak {g}}\)  follows by observing that
the map \(V^\infty \rightarrow C^\infty (G,V), v \mapsto \pi ^v\) 
intertwines the action of \(G\)  with the right translation action on
\(C^\infty (G,V)\) , and this implies that the derived action
corresponds to the action of \({\mathfrak {g}}\)  on \(C^\infty (G,V)\)  by left invariant
vector fields (cf. [106]} for details).
Definition A.1  Let \(G\)  be a Banach–Lie group and
write \(\mathcal {P}(V)\)  for the set of continuous seminorms on \(V\) .
For each \(p \in \mathcal {P}(V)\)  and \(n \in {\mathbb {N}}_0\)  we define a seminorm
\(p_n\)  on \(V^\infty \)  by
\( p_n(v) := \sup \lbrace  p({\tt d}\pi (x_1) \cdots {\tt d}\pi (x_n)v) \colon \Vert x_i\Vert \le 1\rbrace  \) 
and endow \(V^\infty \)  with the locally convex topology defined by these
seminorms.

Theorem A.2  
If \((\pi , V)\)  is a representation of the Banach–Lie group
\(G\)  on the locally convex space \(V\)  defining a continuous action
of \(G\)  on \(V\) , then the
action \(\sigma (g,v) := \pi (g)v\)  of \(G\)  on \(V^\infty \)  is smooth.
If \(V\)  is a Banach space, then \(V^\infty \)  is complete,
i.e., a Fréchet space.

Theorem A.3  
If \((\pi , \mathcal {H})\)  is a unitary representation of a Lie group
\(G\) , then \(v \in \mathcal {H}\)  is a smooth vector if and only if the
corresponding matrix coefficient \(\pi ^{v,v}(g) := \langle \pi (g)v,v\rangle \) 
is smooth on a \(\mathbf {1}\) -neighborhood in \(G\) . If, in addition,
\(v\)  is cyclic, then the representation is smooth.

In the following we write \(\overline{u}\)  for the image of a unitary
operator \(u \in \mathop {\rm U{}}\nolimits (\mathcal {H})\)  in the projective unitary group
\(\mathop {\rm PU{}}\nolimits (\mathcal {H}) :=\mathop {\rm U{}}\nolimits (\mathcal {H})/\) .
Theorem A.4   Let \(G\)  be a connected Lie group,
\(\mathcal {H}\)  a complex Hilbert space and
\(\pi \colon  G \rightarrow \mathop {\rm U{}}\nolimits (\mathcal {H})\)  be a map with \(\pi (\mathbf {1}) = \mathbf {1}\) 
for which the corresponding map
\(\overline{\pi }\colon  G \rightarrow \mathop {\rm PU{}}\nolimits (\mathcal {H})\)  is a group homomorphism.
If there exists a \(v \in \mathcal {H}\)  for which
the function \((g,h) \mapsto \langle \pi (g)\pi (h)v, v\rangle \)  is
smooth on a neighborhood of \((\mathbf {1},\mathbf {1})\)  in \(G \times G\) ,
then the central
extension
\(\widehat{G} := \overline{\pi }^*\mathop {\rm U{}}\nolimits (\mathcal {H}) = \lbrace  (g,u) \in G \times \mathop {\rm U{}}\nolimits (\mathcal {H})\colon  \overline{\pi }(g) = \overline{u} \rbrace  \) 
of \(G\)  by \( is a Lie group and\) v\( is a smooth vector for therepresentation \) (, H)\( of \) G\( by\) (g,u) := u\(.\)
Proof.   
To exhibit \(\widehat{G}\)  as a Lie group, we have to show that
there exists a section \(\sigma \colon  G \rightarrow \widehat{G}\)  for which the
corresponding 2-cocycle
\( f_\sigma (g_1, g_2)= \sigma (g_1) \sigma (g_2) \sigma (g_1 g_2)^{-1} \) 
is smooth in a neighborhood of
\((\mathbf {1},\mathbf {1})\)  ([123]}).
Here we use that \(G\)  is connected.
A particular section \(\sigma \colon  G \rightarrow \widehat{G}\)  is given by
\(\sigma (g) = (g,\pi (g))\) .
Let \(U \subseteq G\)  be an open \(\mathbf {1}\) -neighborhood such that
\( \langle \pi (g)v,\pi (h)^{-1}v \rangle \ne 0 \quad \mbox{ for } \quad g,h \in U. \) 
Its existence follows from our continuity assumption.
If \(U^{\prime }\subseteq U\)  is an open \(\mathbf {1}\) -neighborhood with \(U^{\prime }U^{\prime } \subseteq U\) , we then
have for \(g_1, g_2 \in U^{\prime }\)
\( f_\sigma (g_1, g_2) \langle \pi (g_1 g_2)v, v\rangle = \langle \pi (g_1) \pi (g_2) v, v\rangle , \) 
which leads to
\( f_\sigma (g_1, g_2) = \frac{\langle \pi (g_1)\pi (g_2) v, v\rangle }{\langle \pi (g_1 g_2)v, v\rangle }. \) 
Therefore \(f_\sigma \)  is smooth in a neighborhood of \((\mathbf {1},\mathbf {1})\) .
This shows that \(\widehat{G}\)  is a Lie group and the multiplication map
\(G \times \widehat{G}, (g,t) \mapsto (g, t\pi (g))\) 
is smooth in a neighborhood of \(\mathbf {1}\)  ([123]}).
The representation
\(\widehat{\pi }\)  now satisfies
\( \langle \widehat{\pi }(g,t\pi (g))v, v \rangle = \langle t\pi (g)v, v \rangle = t \langle \pi (g)v,v\rangle , \) 
which is smooth in a neighborhood of \(\mathbf {1}\) . Now
Theorem REF  implies that \(v\)  is a smooth
vector for the representation \(\widehat{\pi }\) .
\(\Box \)
Remark A.5  The assumption that \(G\)  is connected in the preceding theorem
can be removed if \(G\)  is a Banach–Lie group. In this case we assume,
in addition, that \(\overline{\pi }\)  is continuous in the sense that
all functions \(g \mapsto |\langle \pi (g)v,w \rangle |\)  on \(G\)  are continuous
(cf. [125]}).
Then \(\widehat{G}\)  is the pullback of a central extension
of topological groups \(\mathop {\rm U{}}\nolimits (\mathcal {H}) \rightarrow \mathop {\rm PU{}}\nolimits (\mathcal {H})\) , hence in particular
a topological group with respect to the topology inherited from
\(G \times \mathop {\rm U{}}\nolimits (\mathcal {H})\) . Under the assumptions of Theorem REF ,
the central extension \(\widehat{G}_0\)  of the identity component
\(G_0\)  of \(G\)  carries a natural Lie group structure compatible
with the given topology on \(\widehat{G}\) . For each \(g \in \widehat{G}\) , the conjugation
map \(c_g\)  induces a continuous automorphism of the Banach–Lie group
\(\widehat{G}_0\) , and since continuous homomorphisms of Banach–Lie groups
are automatically smooth ([49]}),
\(c_g\)  also defines a Lie automorphism of
\(\widehat{G}_0\) . This implies that \(\widehat{G}\)  carries a unique Lie group
structure which coincides on the open subgroup \(\widehat{G}_0\)  with the given
one (see also [123]}).

Remark A.6  
In the situation of Theorem REF ,
the Lie algebra cocycle defining the central extension
\(\widehat{{\mathfrak {g}}}= \mathop {\bf L{}}\nolimits (\widehat{G})\)  of \({\mathfrak {g}}\)  by \({\mathbb {R}}\)  in the sense that
\( \widehat{{\mathfrak {g}}}= {\mathbb {R}}\oplus _{\eta } {\mathfrak {g}}, \quad [(t,x), (t^{\prime },x^{\prime })] = (\eta (x,x^{\prime }), [x,x^{\prime }]) \) 
can be calculated as follows.

If \({\tt d}\pi \colon  {\mathfrak {g}}\rightarrow \mathop {{\rm End}}\nolimits (\mathcal {H}^\infty )\)  is the map obtained
from the representation \({\tt d}\widehat{\pi }\)  via
\({\tt d}\pi (x) := {\tt d}\widehat{\pi }(0,x)\) , then we have for each unit
vector \(v \in \mathcal {H}^\infty \)  the relation
\( [{\tt d}\pi (x), {\tt d}\pi (y)]v= {\tt d}\pi ([x,y])v + i \eta (x,y)v, \) 
so that
\(\eta (x,y)&= \mathop {{\rm Im}}\nolimits \langle [{\tt d}\pi (x), {\tt d}\pi (y)]v, v\rangle - \mathop {{\rm Im}}\nolimits \langle {\tt d}\pi ([x,y])v, v \rangle \\&= 2\mathop {{\rm Im}}\nolimits \langle {\tt d}\pi (x)v, {\tt d}\pi (y) v\rangle + i \langle {\tt d}\pi ([x,y])v, v \rangle .\) 

The cone of positive definite forms on a Banach space
Let \(V\)  be a Banach space and
\(\mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})\)  be the Banach space of continuous symmetric bilinear
maps \(\beta \colon  V \times V \rightarrow {\mathbb {R}}\) , endowed with the norm
\(\Vert \beta \Vert  := \sup \lbrace  |\beta (v,v)| \colon  \Vert v\Vert  \le 1\rbrace . \) 
Clearly, the set \(\mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})_+\)  of positive semidefinite
bilinear maps is a closed convex cone in \(\mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})\) .
Lemma B.1 

The cone \(\mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})_+\)  has interior points if and only if
\(V\)  is topologically isomorphic to a Hilbert space. If this is
the case, then its interior consists of all those positive definite
forms \(\beta \)  for which the norm \(\Vert v\Vert _\beta := \sqrt{\beta (v,v)}\) 
is equivalent to the norm on \(V\) .

Proof.   
Suppose first that \(V\)  carries a Hilbert space
structure \(\beta \) . Replacing the original norm by an equivalent
Hilbert norm, we may assume that \(V\)  is a real Hilbert space.
Then \(\mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})\)  can be identified with the space
\(\mathop {{\rm Sym}}\nolimits (V)\)  of symmetric operators on \(V\)  by assigning to
\(A \in \mathop {{\rm Sym}}\nolimits (V)\)  the form \(\beta _A(v,v) = \langle Av,w\rangle \) , satisfying
\( |\beta _A(v,v)| = |(Av,v)| \le \Vert A\Vert \Vert v\Vert ^2,\quad \mbox{ so that } \quad \Vert \beta _A\Vert  \le \Vert A\Vert .\) 
The polarization identity
\( (Av,w) = \beta _A(v,w) = \frac{1}{4} (\beta _A(v+w,v+w) -\beta _A(v-w,v-w)) \) 
further implies that \(\Vert A\Vert  \le 2 \beta _A\) , so that the
Banach spaces \(\mathop {{\rm Sym}}\nolimits (V)\)  and \(\mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})\)  are topologically isomorphic.
The identity \(\mathop {{\rm id}}\nolimits _V = \mathbf {1}\)  is an interior point in the cone
of positive operators, \(\mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})_+\)  has interior points.
Suppose, conversely, that \(\beta \in \mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})_+\)  is an
interior point. Then \(\beta - \mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})_+\)  is a 0-neighborhood
in \(\mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})\) , which implies the existence of some
\(c > 0\)  such that \(\Vert \gamma \Vert  \le c\)  implies
\(\gamma (v,v) \le \beta (v,v)\)  for all \(v \in V\) .
Fixing \(v_0 \in V\) , we pick \(\alpha \in V^{\prime }\)  with
\(\Vert \alpha \Vert  = 1\)  and \(\alpha (v_0) = \Vert v_0\Vert \) .
Then \(\gamma (v,w) := c\alpha (v)\alpha (w)\)  is a symmetric bilinear form
with \(\Vert \gamma \Vert  = c\Vert \alpha \Vert ^2 = c\) . We therefore obtain
\( c \Vert v_0\Vert ^2 = \gamma (v_0, v_0) \le \beta (v_0, v_0)\le \Vert \beta \Vert  \Vert v_0\Vert ^2, \) 
showing that \(\beta \)  is positive definite and that the norm
\(\sqrt{\beta (v,v)}\)  is equivalent to the norm on \(V\) .
\(\Box \)
Remark B.2  
Let \((V,\omega )\)  be a weakly symplectic Banach space, i.e.,
\(\omega \)  is a non-degenerate skew-symmetric bilinear form.
Then
\( \mathfrak {sp}(V,\omega ) := \lbrace  X \in B(V) \colon  (\forall v,w\in V)\,\omega (Xv,w) + \omega (v,Xw) = 0\rbrace  \) 
is the corresponding symplectic Lie algebra. In particular,
\(X \in B(V)\)  belongs to \(\mathfrak {sp}(V,\omega )\)  if and only if
the bilinear form \(\omega (Xv,w)\)  is symmetric. The corresponding
quadratic function \(H_X(v) := {\textstyle {\frac{1}{2}}}\omega (Xv,v)\)  is called the
Hamiltonian function defined by \(X\).
If there exists an \(X \in \mathfrak {sp}(V,\omega )\)  for which
\((v,w) := \omega (Xv,w)\)  defines a Hilbert space structure
on \(V\) , then each continuous linear
functional \(\alpha \in V^{\prime }\)  is of the form \(i_v\omega \)  for
some \(v \in V\) . This means that
\((V,\omega )\)  is strongly symplectic,
i.e., the map
\(\Phi _\omega \colon  V \rightarrow V^{\prime }, v \mapsto i_v\omega \)  is surjective,
hence a topological isomorphism by the Open Mapping Theorem.
We further see that \(X\)  is injective. To see that it is also surjective,
let \(u \in V\)  and represent the continuous linear functional
\(i_u\omega \)  as \((w,\cdot )\)  for some \(w \in V\) . Then
\(\omega (u,v) = (w,v) = \omega (Xw,v)\)  for all \(v \in V\) ,
and thus \(Xw = u\) . Hence \(X\)  is a topological isomorphism,
satisfying \(\omega (u,v) = (X^{-1}u,v)\)  for \(v,w \in V\) . This further implies
that \(X\)  is skew-symmetric.

Proposition B.3  
Let \((V,\omega )\)  be a strongly symplectic Banach space.
Then the convex cone \(\lbrace  X \in \mathfrak {sp}(V,\omega ) \colon  H_X \ge 0\rbrace \)  has interior
points if and only if \(V\)  is topologically isomorphic to a
Hilbert space. If this is the case,
then there exists a complex structure \(I \in \mathfrak {sp}(V,\omega )\)  for
which \(H_I\)  defines a compatible Hilbert space structure on \(V\) , i.e.,
\( \omega (v,w) = \mathop {{\rm Im}}\nolimits \langle v, w\rangle \) 
for the underlying complex Hilbert space.

Proof.   
(cf. [6]})
If \(\beta \colon  V \times V \rightarrow {\mathbb {R}}\)  is a symmetric bilinear form,
then \(\Phi _\beta \colon  V \rightarrow V^{\prime }, v \mapsto i_v\beta \)  also is a continuous
linear map, and
\(B := \Phi _\omega ^{-1} \circ \Phi _\beta \colon  V \rightarrow V\)  is continuous
linear with
\( \omega (Bv,v) = \Phi _\omega (Bv)(v)= \Phi _\beta (v)(v) = \beta (v,v), \) 
so that every symmetric bilinear form can be represented by an
element of \(\mathfrak {sp}(V,\omega )\) , and we obtain a topological isomorphism
\(\mathfrak {sp}(V,\omega ) \cong \mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})\) . Combining this with
Lemma REF  proves the first assertion.
We now assume that \(V\)  is a real Hilbert space with the scalar product
\((v,w) = \omega (Xv,w)\) , where \(X\)  is as above. Then \(A := X^{-1}\)  is an
invertible skew-symmetric operator with \(\omega (v,w) = (Av,w)\) .
Then the complex linear extension \(A_ to the complex Hilbertspace \) V yields a self-adjoint operator \(iA_.The complex conjugation \)\( of \) V with respect to \(V\) 
now satisfies \(\sigma \circ iA_\sigma = - i A_, sothat \) Spec(iA\( is a compact symmetric subset of \) R\(, notcontaining \) 0\(. We therefore have an orthogonal decomposition\) V V+ V-\( into the positive and negative spectralsubspaces of \) iA. Since \(\sigma (V_\pm ) = V_\mp \) , we obtain
an isomorphism \(V \cong V_+\) , and hence a complex structure
\(I\)  on \(V\) , corresponding to multiplication by \(i\)  on \(V_+\) .
This means that \(v_+ \in V_+\)  with \(v = v_+ + \sigma (v_+)\) 
satisfies
\(\omega (Iv,v)&= (AIv,v)= (AI(v_+ + \sigma (v_+)), v_+ + \sigma (v_+)) \\&= (A_i v_+ - i \sigma (v_+)), v_+ + \sigma (v_+))= i (A_v_+ - \sigma (v_+)), v_+ + \sigma (v_+)) \\&= i \big ((A_v̏_+, v_+) - (v_+, A_v̏_+)\big )= 2 (iA_v̏_+, v_+),\) 

so we obtain a complex structure \(I\)  on \(V\) 
for which the Hamiltonian \(H_I\)  defines a Hilbert structure on
\(V\) . Since \(I\)  is skew-symmetric, \((V,I)\)  inherits the structure
of a complex Hilbert space with respect to the scalar product
\( \langle v, w \rangle := \omega (Iv,w) + i \omega (Iv,Iw) = \omega (Iv,w) + i \omega (v,w). \) 
Therefore our assumption leads to the representation
of \(\omega \)  as \(\omega (v,w) = \mathop {{\rm Im}}\nolimits \langle v, w \rangle \)  for a complex
Hilbert space structure on \(V\) .
\(\Box \)
Involutive Lie algebras with root decomposition
Definition C.1   (a) We call an abelian subalgebra
\({\mathfrak {t}}\)  of the real Lie algebra \({\mathfrak {g}}\)  a compactly embedded
Cartan subalgebra if \({\mathfrak {t}}\)  is maximal abelian and \(\mathop {{\rm ad}}\nolimits {\mathfrak {t}}\) 
is simultaneously diagonalizable on the complexification \({\mathfrak {g}}_with purely imaginary eigenvalues.Then we have a {\it root decomposition}\)\( {\mathfrak {g}}_ {\mathfrak {t}}_ \sum _{\alpha \in \Delta } {\mathfrak {g}}_\alpha , \)\(where \) g= { x g (h t [h,x]= (h)
x}\( and\)\(\Delta := \lbrace  \alpha \in {\mathfrak {t}}_* \setminus \lbrace 0\rbrace  \colon  {\mathfrak {g}}_\alpha \ne  \lbrace 0\rbrace \rbrace \)\( is the corresponding {\it root system}.\) If \(\sigma \colon  {\mathfrak {g}}_{\mathfrak {g}}_ denotes the complex conjugation withrespect to \) g\(, we write \) x* := -(x)\( for \) x g, so that
\({\mathfrak {g}}= \lbrace  x \in {\mathfrak {g}}_ x^* = -x\rbrace \) .
We then have
(I1)
\(\alpha (x) \in {\mathbb {R}}\)  for \(x \in i {\mathfrak {t}}\) .
(I2)
\(\sigma ({\mathfrak {g}}_\alpha ) = {\mathfrak {g}}_{-\alpha }\)  for \(\alpha \in \Delta .\)


Lemma C.2 
 For \(0 \ne  x_{\alpha } \in {\mathfrak {g}}_{\alpha }\)  the subalgebra
\({\mathfrak {g}}_x_\alpha ) := \mathop {{\rm span}}\nolimits \lbrace x_\alpha , x_{\alpha }^*,[x_\alpha , x_{\alpha }^*]\rbrace \) 
is \(\sigma \) -invariant and of one of the following types:
(A)
The abelian type: \([x_\alpha , x_{\alpha }^*] = 0\) , i.e.,
\({\mathfrak {g}}_x_\alpha )\)  is two dimensional abelian.
(N)
The nilpotent type: \([x_\alpha , x_{\alpha }^*] \ne  0\) 
and \(\alpha ([x_\alpha , x_{\alpha }^*]) = 0\) , i.e.,
\({\mathfrak {g}}_x_\alpha )\)  is a three dimensional Heisenberg algebra.
(S)
The simple type: \(\alpha ([x_\alpha , x_{\alpha }^*]) \ne  0\) ,
i.e., \({\mathfrak {g}}_x_\alpha ) \cong \mathop {{\mathfrak {sl} }}\nolimits _2(\) . In this case we distinguish the
two cases:
(CS)
\(\alpha ([x_\alpha , x_{\alpha }^*]) > 0\) , i.e.,
\({\mathfrak {g}}_x_\alpha ) \cap {\mathfrak {g}}\cong \mathop {{\mathfrak {su} }}\nolimits _2(\) , and
(NS)
\(\alpha ([x_\alpha , x_{\alpha }^*]) < 0\) , i.e.,
\({\mathfrak {g}}_x_\alpha ) \cap {\mathfrak {g}}\cong \mathop {{\mathfrak {su} }}\nolimits _{1,1}( \cong \mathop {{\mathfrak {sl} }}\nolimits _2({\mathbb {R}})\) .


Proof.   
First we note that, in view of \(x_\alpha ^* \in {\mathfrak {g}}_{-\alpha }\) ,
[87]} applies, and we see that \({\mathfrak {g}}_x_\alpha )\)  is of one of
the three types (A), (N) and (S). We note that
\(\alpha ([x_\alpha , x_\alpha ^*]) \in {\mathbb {R}}\) 
because of (I2) and \([x_\alpha , x_\alpha ^*] \in i{\mathfrak {t}}\) .
Now it is easy to check that
\({\mathfrak {g}}_x_\alpha ) \cap {\mathfrak {g}}\)  is of type (CS), resp., (NS), according to the sign
of this number.
\(\Box \)
The following proposition provides useful information for the
analysis of invariant cones and orbit projections.
Here we write \(p_{\mathfrak {t}}\colon  {\mathfrak {g}}\rightarrow {\mathfrak {t}}\)  for the
projection along \([{\mathfrak {t}},{\mathfrak {g}}] = \sum _{\alpha } ({\mathfrak {g}}_\alpha + {\mathfrak {g}}_{-\alpha }) \cap {\mathfrak {g}}\) ,
and \(p_{{\mathfrak {t}}^*} \colon  {\mathfrak {g}}^* \rightarrow {\mathfrak {t}}^*\)  for the restriction map.
Proposition C.3 

Let \(x \in {\mathfrak {t}}\) , \(x_\alpha \in {\mathfrak {g}}_\alpha \)  and
\(\lambda \in {\mathfrak {t}}^*\) . Then the following assertions hold:
(i)
\(p_{\mathfrak {t}}(e^{{\mathbb {R}}\mathop {{\rm ad}}\nolimits (x_\alpha - x_\alpha ^*)}x)= x +{\left\lbrace \begin{array}{ll}{\mathbb {R}}^+ \alpha (x) [x_\alpha ^*, x_\alpha ] & \text{ for }\ \alpha ([x_\alpha , x_\alpha ^*]) \le 0 \\[0,2] { \frac{\alpha (x)}{\alpha ([x_\alpha , x_\alpha ^*])}}[x_\alpha ^*, x_\alpha ] & \text{ for }\ \alpha ([x_\alpha , x_\alpha ^*]) > 0.\end{array}\right.}\)
(ii)
\(p_{{\mathfrak {t}}^*}(e^{{\mathbb {R}}\mathop {{\rm ad}}\nolimits ^* (x_\alpha - x_\alpha ^*)}\lambda )= \lambda +{\left\lbrace \begin{array}{ll}{\mathbb {R}}^+ \lambda ([x_\alpha ^*, x_\alpha ]) \alpha & \text{ for }\ \alpha ([x_\alpha , x_\alpha ^*]) \le 0 \\[0,2] \frac{\lambda ([x_\alpha ^*, x_\alpha ])}{\alpha ([x_\alpha , x_\alpha ^*])}\alpha & \text{ for }\ \alpha ([x_\alpha , x_\alpha ^*]) > 0.\end{array}\right.}\)


Proof.   
(i) is an immediate consequence of [1]},
and (ii) follows from (i) and the relation
\(p_{{\mathfrak {t}}^*}(\lambda )(e^{\mathop {{\rm ad}}\nolimits y}x)= \lambda (p_{\mathfrak {t}}(e^{\mathop {{\rm ad}}\nolimits y}x)).\) 
\(\Box \)
Some facts on Fock spaces
Let \(\mathcal {H}\)  be a complex Hilbert space. We endow the \(n\) -fold tensor
product with the canonical Hilbert structure defined by
\( \langle v_1 \otimes \cdots \otimes v_n, w_1 \otimes \cdots \otimes w_n \rangle := \prod _{j = 1}^n \langle v_j, w_j \rangle , \) 
and form the Hilbert space direct sum
\(T(\mathcal {H}) := \widehat{\bigoplus }_{n \in {\mathbb {N}}_0} \mathcal {H}^{\otimes n}\) .
In \(\mathcal {H}^{\otimes n}\)  we write \(S^n(\mathcal {H})\)  for the closed subspace
generated by the symmetric tensors and
\(\Lambda ^n(\mathcal {H})\)  for the closed subspace generated by the
alternating tensors. We thus obtain subspaces
\( S(\mathcal {H}) := \widehat{\bigoplus }_{n \in {\mathbb {N}}_0} S^n(\mathcal {H})\quad \mbox{ and } \quad \Lambda (\mathcal {H}) := \widehat{\bigoplus }_{n \in {\mathbb {N}}_0} \Lambda ^n(\mathcal {H}) \) 
of \(T(\mathcal {H})\)  and write \(P_s\) , resp., \(P_a\)  for the corresponding
orthogonal projections:
\( P_s(f_1 \otimes \cdots \otimes f_n)= \frac{1}{n!} \sum _{\sigma \in S_n} f_{\sigma (1)} \otimes \cdots \otimes f_{\sigma (n)} \) 
and
\( P_a(f_1 \otimes \cdots \otimes f_n)= \frac{1}{n!} \sum _{\sigma \in S_n} \mathop {{\rm sgn}}\nolimits (\sigma )f_{\sigma (1)} \otimes \cdots \otimes f_{\sigma (n)}. \) 
The dense subspace \(S(\mathcal {H})_0 := \sum _{n \ge 0} S^n(\mathcal {H})\)  of \(S(\mathcal {H})\) 
carries an associative
algebra structure, given by
\( f_1 \vee \cdots \vee f_n := \sqrt{n!}P_s(f_1 \otimes \cdots \otimes f_n) \) 
and likewise \(\Lambda (\mathcal {H})_0 := \sum _{n \ge 0} \Lambda ^n(\mathcal {H})\) 
inherits an algebra structure defined by
\( f_1 \wedge \cdots \wedge f_n := \sqrt{n!}P_a(f_1 \otimes \cdots \otimes f_n). \) 
A unit vector \(\Omega \)  in the one-dimensional space
\(S^0(\mathcal {H}) = \Lambda ^0(\mathcal {H})\)  is called a vacuum vector.
Lemma D.1 
 We have
\(T \vee S= \sqrt{{n+m \atopwithdelims ()n}} P_s(T \otimes S) \quad \mbox{ for } \quad T \in S^n(\mathcal {H}), S \in S^m(\mathcal {H})\) 
and
\(T \wedge S= \sqrt{{n+m \atopwithdelims ()n}} P_a(T \otimes S) \quad \mbox{ for } \quad T \in \Lambda ^n(\mathcal {H}), S \in \Lambda ^m(\mathcal {H}).\) 


Proof.   
For the symmetric case we first note that
\(f^n = \sqrt{n!} f^{\otimes n}\) , so that
\( f^n \vee g^m= \sqrt{(n+m)!} P_s(f^{\otimes n} \otimes g^{\otimes m})= \sqrt{{n+m \atopwithdelims ()n}} P_s(f^n \otimes g^m). \) 
Since the elements \(f^n\)  generate \(S^n(\mathcal {H})\)  topologically,
(REF ) follows.
For the alternating case we obtain for
\(T = f_1 \wedge \cdots \wedge f_n\)  and
\(S = g_1 \wedge \cdots \wedge g_m\)  the relation
\(&\ \ \ \ \frac{1}{\sqrt{(n+m)!}}T \wedge S= P_a(f_1 \otimes \cdots \otimes g_m)\\&= P_a(P_a(f_1 \otimes \cdots \otimes f_n) \otimes P_a(g_1 \otimes \cdots \otimes g_m))= \frac{1}{\sqrt{n!m!}} P_a(T \otimes S).\) 

\(\Box \)
Remark D.2  
(a) For the norms of the product of \(T \in S^n(\mathcal {H})\)  and \(S \in S^m(\mathcal {H})\) ,
we obtain with (REF )
\( \frac{1}{\sqrt{(n+m)!}} \Vert T \vee S\Vert = \frac{1}{\sqrt{n!}\sqrt{m!}} \Vert P_s(T \otimes S)\Vert \le \frac{1}{\sqrt{n!}\sqrt{m!}} \Vert T \otimes S\Vert = \frac{\Vert T\Vert }{\sqrt{n!}} \frac{\Vert S\Vert }{\sqrt{m!}}.\) 
From that we derive for \(T \in S^k(\mathcal {H})\)  and \(n \in {\mathbb {N}}\)  the relation
\( \frac{1}{\sqrt{(kn)!}} \Vert T^n\Vert \le \Big (\frac{1}{\sqrt{k!}}\Vert T\Vert \Big )^n, \) 
and for \(T \in S^2(\mathcal {H})\)  we find in particular
\(\Vert e^T\Vert ^2= \sum _{n = 0}^\infty \frac{1}{(n!)^2} \Vert T^n\Vert ^2\le \sum _{n = 0}^\infty \frac{(2n)!}{(n!)^2 2^n} \Vert T\Vert ^{2n},\) 
which converges for \(\Vert T\Vert ^2 < \frac{1}{2}\) .
(b) For \(T \in \Lambda ^n(\mathcal {H})\)  and \(S \in \Lambda ^m(\mathcal {H})\) 
we likewise obtain with (REF )
\( \frac{1}{\sqrt{(n+m)!}} \Vert T \wedge S\Vert \le \frac{\Vert T\Vert }{\sqrt{n!}} \frac{\Vert S\Vert }{\sqrt{m!}},\) 
which leads for \(T \in \Lambda ^2(\mathcal {H})\)  to
\(\Vert e^T\Vert ^2= \sum _{n = 0}^\infty \frac{1}{(n!)^2} \Vert T^n\Vert ^2\le \sum _{n = 0}^\infty \frac{(2n)!}{(n!)^2 2^n} \Vert T\Vert ^{2n}.\) 

Lemma D.3  
Let \(A\)  be an antilinear Hilbert–Schmidt operator on \(\mathcal {H}\) 
and define \(A^*\)  by \(\langle A^*v,w \rangle = \langle Aw,v\rangle \) .
If \(A^* = A\) , then there exists a unique element
\(\widehat{A} \in S^2(\mathcal {H})\)  with
\( \langle \widehat{A}, f_1 \vee f_2 \rangle = \langle Af_1, f_2 \rangle f\quad \mbox{ for } \quad f_1, f_2 \in \mathcal {H}, \) 
and if \(A^* = - A\) , there exists a unique element \(\widehat{A} \in \Lambda ^2(\mathcal {H})\) 
with
\( \langle \widehat{A}, f_1 \wedge f_2 \rangle = \langle Af_1, f_2 \rangle \quad \mbox{ for } \quad f_1, f_2 \in \mathcal {H}. \) 
Moreover, the following assertions hold:
(i)
\(\Vert \widehat{A}\Vert ^2 = {\textstyle {\frac{1}{2}}}\Vert A\Vert _2^2\) .
(ii)
If \(A\)  and \(B\)  are hermitian and antilinear, then
\(\langle \widehat{A}, \widehat{B} \rangle = {\textstyle {\frac{1}{2}}}\mathop {{\rm tr}}\nolimits (AB)\) .
(iii)
If \(A\)  and \(B\)  are skew-hermitian and antilinear, then
\(\langle \widehat{A}, \widehat{B} \rangle = -{\textstyle {\frac{1}{2}}}\mathop {{\rm tr}}\nolimits (AB)\) .


Proof.   
(a) First we consider the case where \(A^* = A\) .
Let \((e_j)_{j \in J}\)  be an orthonormal basis of \(\mathcal {H}\) .
Then we have in \(S^2(\mathcal {H})\)  the relations
\( \Vert e_j^2\Vert ^2 = 2\Vert  e_j \otimes e_j\Vert ^2 = 2\quad \mbox{ and } \quad \Vert e_j \vee e_k\Vert ^2= 2\Vert {\textstyle {\frac{1}{2}}}(e_j \otimes e_k + e_k \otimes e_j)\Vert ^2 = 1, \quad j\ne k. \) 
If \(<\)  denotes a linear order on \(J\) , we thus obtain the
orthonormal basis
\(\frac{1}{\sqrt{2}} e_j^2, \quad e_j \vee e_k, \quad j < k, \quad \mbox{ of } \quad S^2(\mathcal {H}).\) 
This leads to
\(\Vert \widehat{A}\Vert ^2&= \sum _j {\textstyle {\frac{1}{2}}}|\langle \widehat{A}, e_j^2 \rangle |^2+ \sum _{j<k} |\langle \widehat{A}, e_j \vee e_k \rangle |^2 \\&= \sum _j {\textstyle {\frac{1}{2}}}|\langle Ae_j, e_j \rangle |^2+ \sum _{j<k} |\langle Ae_j, e_k \rangle |^2= {\textstyle {\frac{1}{2}}}\Vert A\Vert _2^2.\) 
For \(A^* = -A\)  we similarly get
\(\Vert \widehat{A}\Vert ^2&= \sum _{j<k} |\langle \widehat{A}, e_j \vee e_k \rangle |^2= \sum _{j<k} |\langle Ae_j, e_k \rangle |^2 = {\textstyle {\frac{1}{2}}}\Vert A\Vert _2^2.\) 
(b) Let \(\mathop {{\rm Herm}}\nolimits _2(\mathcal {H})_a\)  denote the complex subspace
of hermitian antilinear
Hilbert–Schmidt operators.
Then the prescription \(\langle A, B \rangle := \mathop {{\rm tr}}\nolimits (AB) = \mathop {{\rm tr}}\nolimits (AB^*)\)  defines a
sesquilinear form on this space with
\( \overline{\mathop {{\rm tr}}\nolimits (AB)} = \mathop {{\rm tr}}\nolimits ((AB)^*) = \mathop {{\rm tr}}\nolimits (B^*A^*) = \mathop {{\rm tr}}\nolimits (BA), \) 
so that it is hermitian. Its restriction to the diagonal
satisfies
\(\langle A,A \rangle = \mathop {{\rm tr}}\nolimits (A^2)= \sum _{j \in J} \langle A^2e_j, e_j \rangle = \sum _{j \in J} \langle Ae_j, Ae_j \rangle = \Vert A\Vert _2^2. \) 
In view of (a), polarization implies that
\(\mathop {{\rm tr}}\nolimits (AB) = \langle A, B \rangle = 2 \langle \widehat{A}, \widehat{B}\rangle \) 
for \(A, B \in \mathop {{\rm Herm}}\nolimits _2(\mathcal {H})_a\) .
(c) For the space \(\mathop {{\rm Aherm}}\nolimits _2(\mathcal {H})_a\)  of skew-hermitian antilinear
Hilbert–Schmidt operators the same argument works with
\(\langle A, B \rangle := -\mathop {{\rm tr}}\nolimits (AB) = \mathop {{\rm tr}}\nolimits (AB^*)\) .
\(\Box \)
